The speed of joint movement with the organization of God. Solving joint movement problems How to find the speed of joint movement

So let's say our bodies are moving in the same direction. How many cases do you think there can be for such a condition? That's right, two.

Why does this happen? I am sure that after all the examples, you can easily figure out how to display these formulas.

Understood? Well done! It's time to solve the problem.

The fourth task

Kolya is driving to work at a speed of km / h. Kolya's colleague Vova is driving at a speed of km / h. Kolya lives from Vova at a distance of km.

How long will it take for Vova to catch up with Kolya if they left the house at the same time?

Have you counted? Let's compare the answers - I managed that Vova will catch up with Kolya in an hour or a minute.

Let's compare our solutions ...

The picture looks like this:

Does it look like yours? Well done!

Since the problem asks how long the guys met, and they left at the same time, the time they traveled will be the same, as well as the meeting place (in the figure it is indicated by a dot). Composing equations, let's take time for.

So, Vova made his way to the meeting place. Kolya made his way to the meeting place. It's clear. Now we are dealing with the axis of movement.

Let's start with the path taken by Kolya. Its path () is shown as a segment in the figure. And what does Vova's path () consist of? That's right, from the sum of the segments and, where is the initial distance between the guys, and is equal to the path that Kolya made.

Based on these conclusions, we get the equation:

Understood? If not, just read this equation again and look at the points marked on the axis. Drawing helps, doesn't it?

hours or minutes minutes.

I hope this example gives you an idea of ​​how important well-composed drawing!

And we are smoothly moving on, or rather, have already passed to the next point of our algorithm - bringing all values ​​to the same dimension.

The rule of three "R" - dimension, rationality, calculation.

Dimension.

It is far from always that the tasks give the same dimension for each participant in the movement (as it was in our easy tasks).

For example, you can find tasks where it is said that the bodies moved for a certain number of minutes, and their speed is indicated in km / h.

We cannot just take and substitute values ​​in the formula - the answer will be wrong. Even in units of measurement, our answer "will not pass" the reasonableness test. Compare:

See? With correct multiplication, we also reduce the units of measurement, and, accordingly, a reasonable and correct result is obtained.

What happens if we do not translate into one measurement system? Strange dimension of the answer and% incorrect result.

So, let me remind you, just in case, the values ​​of the basic units of measurement of length and time.

Length units:

centimeter = millimeters

decimeter = centimeters = millimeters

meter = decimetres = centimeters = millimeters

kilometer = meters

Time units:

minute = seconds

hour = minutes = seconds

day = hours = minutes = seconds

Advice: When converting time units (minutes to hours, hours to seconds, etc.), imagine a clock face in your head. The naked eye can see that the minutes are a quarter of the dial, i.e. hours, minutes is a third of the dial, i.e. hours, and a minute is an hour.

And now a very simple task:

Masha rode her bike from home to the village at a speed of km / h for minutes. What is the distance between the car's house and the village?

Have you counted? The correct answer is km.

minutes is an hour, and more minutes from an hour (I mentally imagined a clock face and said that minutes is a quarter of an hour), respectively - min = h.

Reasonableness.

Do you understand that the speed of a car cannot be km / h, unless, of course, we are talking about a sports car? And even more so, it cannot be negative, right? So, rationality, that's about it)

Payment.

See if your solution "passes" for dimension and rationality, and only then check the calculations. It is logical - if there is an inconsistency with dimension and rationality, then it is easier to cross everything out and start looking for logical and mathematical errors.

"Love for tables" or "when drawing is not enough"

Movement problems are not always as simple as we solved earlier. Very often, in order to correctly solve a problem, you need not just draw a competent drawing, but also draw up a table with all the conditions given to us.

First task

From point to point, the distance between which is km, a cyclist and a motorcyclist left at the same time. It is known that a motorcyclist travels more kilometers per hour than a cyclist.

Determine the speed of the cyclist if it is known that he arrived at the point minutes later than the motorcyclist.

Here is such a task. Pull yourself together and read it several times. Have you read it? Start drawing - a straight line, point, point, two arrows ...

In general, draw, and now let's compare what you got.

It's kind of empty, isn't it? We draw a table.

As you remember, all movement tasks consist of components: speed, time and path... It is these graphs that any table will consist of in such tasks.

However, we will add one more column - name about whom we write information - motorcyclist and cyclist.

Also indicate in the cap dimension, in which you will enter the values ​​there. You remember how important this is, right?

Have you got a table like this?

Now let's analyze everything that we have, and in parallel enter the data into the table and into the figure.

The first thing we have is the path that the cyclist and motorcyclist have done. It is the same and equal to km. We bring in!

Take the speed of the cyclist as, then the speed of the motorcyclist will be ...

If the solution to the problem does not work with such a variable, it's okay, we'll take another one until we reach the victorious one. It happens, the main thing is not to be nervous!

The table has changed. We have only one column left unfilled - time. How to find the time when there is a path and speed?

That's right, divide the path into speed. Put it on the table.

So our table has been filled in, now you can enter the data on the figure.

What can we reflect on it?

Well done. The speed of movement of the motorcyclist and cyclist.

Let's re-read the problem again, look at the figure and the completed table.

What data is not reflected in either the table or the figure?

Right. The time at which the motorcyclist arrived earlier than the cyclist. We know that the difference in time is minutes.

What should we do next? That's right, translate the time given to us from minutes to hours, because the speed is given to us in km / h.

The magic of formulas: drawing up and solving equations is manipulation that leads to the only correct answer.

So, as you already guessed, now we will make up the equation.

Equation drawing:

Take a look at your table, at the last condition that was not included in it and think, the relationship between what and what can we take into the equation?

Right. We can make an equation based on the time difference!

Is it logical? The cyclist rode more, if we subtract the motorcyclist's travel time from his time, we just get the difference given to us.

This equation is rational. If you do not know what it is, read the topic "".

We bring the terms to a common denominator:

Let's open the brackets and give similar terms: Ugh! Got it? Try your hand at the next challenge.

Solution of the equation:

From this equation we get the following:

Let's open the brackets and move everything to the left side of the equation:

Voila! We have a simple quadratic equation. We decide!

We received two options for an answer. See what we got for? That's right, the speed of the cyclist.

We recall the rule "3P", more specifically "rationality". Do you understand what I mean? Exactly! The speed cannot be negative, therefore our answer is km / h.

Second task

Two cyclists set off for a mile-long run at the same time. The first was driving at a speed that is km / h higher than the speed of the second, and arrived at the finish line hours earlier than the second. Find the speed of the cyclist who finished second. Give your answer in km / h.

I remind the solution algorithm:

• Read the problem a couple of times - learn all the details. Got it?
• Start drawing a drawing - in which direction are they moving? how far did they go? Drew?
• Check if all your quantities are of the same dimension and start writing out briefly the condition of the problem, drawing up a table (do you remember what graphs are there?).
• While you are writing all this, think about what to take for? Have you chosen? Write it down in the table! Well, now it's simple: we make an equation and solve it. Yes, and finally - remember about "3P"!
• I've done everything? Well done! It turned out that the speed of the cyclist is km / h.

-"What color is your car?" - "She's beautiful!" Correct answers to the questions posed

Let's continue our conversation. So what is the speed of the first cyclist? km / h? I really hope that you are not nodding in the affirmative right now!

Carefully read the question: “What is the speed of the first a cyclist? "

Do you understand what I mean?

Exactly! Received is not always the answer to the question posed!

Read the questions thoughtfully - perhaps after finding it, you will need to perform some more manipulations, for example, add km / h, as in our task.

Another point - often in tasks everything is indicated in hours, and the answer is asked to be expressed in minutes, or all the data is given in km, and the answer is asked to be written in meters.

Watch the dimension not only during the solution itself, but also when you write down the answers.

Circular tasks

Bodies in tasks may not necessarily move straight, but also in a circle, for example, cyclists can ride on a circular track. Let us examine such a problem.

Problem number 1

A cyclist left the point of the circular route. In minutes he had not yet returned to the point and a motorcyclist followed him from the point. Minutes after departure, he caught up with the cyclist for the first time, and minutes after that he caught up with him a second time.

Find the speed of the cyclist if the length of the track is km. Give your answer in km / h.

Solution to problem number 1

Try to draw a picture for this problem and fill in the table for it. Here's what I got:

Between the meetings, the cyclist traveled the distance, and the motorcyclist -.

But at the same time, the motorcyclist drove exactly one more lap, this can be seen from the figure:

I hope you understand that they didn't actually go in a spiral - the spiral just schematically shows that they go in a circle, passing the same points of the track several times.

Understood? Try to solve the following tasks yourself:

Tasks for independent work:

1. Two mo-to-tsik-li-a hundred start-to-eut one-time-but-in-one-right-ley out of two dia-metral-but pro-ti-in-po false points of a circular route, the length of which is equal to km. After how many minutes, mo-to-cycl-lis-sts will equalize for the first time, if the speed of one of them is greater than the speed of the other by km / h ho-ho?
2. From one point on a steep track, the length of which is equal to km, one-n-time-but in one on-the-right-ley there are two motorcyclists. The speed of the first motorcycle is equal to km / h, and minutes after the start, he operated the second motorcycle for one lap. Nai-di-te speed of the second-ro-th motorcycle. Give your answer in km / h.

Solving problems for independent work:

1. Let km / h be the speed of the first mo-to-cycle-leaf, then the speed of the second mo-to-cycle-leaf is equal to km / h. Let the first time my-that-cycl-lis-sts will be equal in hours. In order for the mo-to-tsik-lis-sts to be equal, the faster one must overcome from-chal-but raz-de-la-yu-them distance, equal to lo-vi-not the length of the route.

   We get that the time is equal to hours = minutes.

2. Let the speed of the second motorcycle be equal to km / h. In an hour, the first motorcycle traveled more kilometers than the second, respectively, we get the equation:

   The speed of the second rider is km / h.

Tasks for the course

Now that you are excellent at solving problems "on land", let's go into the water and look at the daunting problems associated with the current.

Imagine that you have a raft and you lowered it into the lake. What's going on with him? Right. It stands because a lake, a pond, a puddle, after all, is stagnant water.

The speed of the current in the lake is .

The raft will only go if you start rowing yourself. The speed that he gains will be own speed of the raft. It doesn't matter where you sail - to the left, to the right, the raft will move as fast as you paddle. It's clear? It is logical.

Now imagine that you are lowering the raft onto the river, turning away to take the rope ..., turning, and he ... swam away ...

This is because the river has a current speed, which carries your raft in the direction of the current.

At the same time, its speed is equal to zero (you are standing in shock on the shore and do not row) - it moves with the speed of the current.

Understood?

Then answer this question - "How fast will the raft float on the river if you are sitting and rowing?" Thinking?

There are two possibilities here.

Option 1 - you go with the flow.

And then you swim at your own speed + current speed. The flow, as it were, helps you to move forward.

2nd option - t You are swimming against the tide.

Hard? Correct, because the current is trying to "throw" you back. You make more and more effort to swim at least meters, respectively, the speed with which you move is equal to your own speed - the speed of the current.

Let's say you need to swim km. When will you cover this distance faster? When will you go with the flow or against?

Let's solve the problem and check it out.

Add to our route data on the speed of the current - km / h and on the own speed of the raft - km / h. How much time will you spend moving with and against the flow?

Of course, you easily coped with this task! Downstream - an hour, and upstream as much as an hour!

This is the whole essence of the tasks for movement with the flow.

Let's complicate the task a little.

Problem number 1

The boat with a motor sailed from point to point in an hour, and back - in an hour.

Find the current speed if the boat speed in still water is km / h

Solution to problem number 1

Let's denote the distance between points as, and the speed of the current as.

	Path S	Speed ​​v, km / h	Time t, hours
A -> B (upstream)			3
B -> A (downstream)			2

We see that the boat travels the same path, respectively:

What did we take for?

Current speed. Then this will be the answer :)

The current speed is equal to km / h.

Problem number 2

The kayak went from point to point located in km from. After staying at the point for an hour, the kayak went back and returned to point c.

Determine (in km / h) your own speed of the kayak if you know that the speed of the river is km / h.

Solution to problem number 2

So let's get started. Read the problem several times and draw a drawing. I think you can easily solve this on your own.

Are all values ​​expressed in one form? No. The rest time is indicated in both hours and minutes.

Let's translate this into hours:

hour minutes = h.

Now all values ​​are expressed in one form. Let's start filling out the table and finding what we will take for.

Let be the kayak's own speed. Then, the speed of the kayak downstream is equal, and upstream is equal.

Let's write this data, as well as the path (it is, as you understand, the same) and the time, expressed in terms of the path and speed, in a table:

	Path S	Speed ​​v, km / h	Time t, hours
Against the stream	26
With the flow	26

Let's calculate how much time the kayak has spent on its journey:

Did she swim all the hours? We reread the problem.

No, not all. She had a rest of an hour minutes, respectively, from the hours we subtract the rest time, which, we have already converted into hours:

h the kayak really floated.

Let us bring all the terms to a common denominator:

Let's expand the brackets and present similar terms. Next, we solve the resulting quadratic equation.

With this, I think you can handle it yourself. What answer did you get? I have km / h.

Let's sum up

ADVANCED LEVEL

Movement tasks. Examples of

Consider examples with solutionsfor each type of task.

Movement with the flow

Some of the simplest tasks are - river driving tasks... Their whole point is as follows:

• if we move with the current, the speed of the current is added to our speed;
• if we move against the current, the current velocity is subtracted from our speed.

Example # 1:

The boat sailed from point A to point B for hours and back - for hours. Find the current speed if the speed of the boat in still water is km / h.

Solution # 1:

Let's denote the distance between points as AB, and the speed of the current as.

We will enter all data from the condition into the table:

	Path S	Speed ​​v, km / h	Time t, hours
A -> B (upstream)	AB	50-x	5
B -> A (downstream)	AB	50 + x	3

For each row of this table, you need to write the formula:

In fact, you don't have to write equations for each row in the table. After all, we see that the distance traveled by the boat back and forth is the same.

This means that we can equate the distance. To do this, use immediately the formula for the distance:

You often have to use and formula for time:

Example # 2:

Against the current, the boat sails a distance in km for an hour longer than downstream. Find the speed of the boat in still water if the current speed is km / h.

Solution # 2:

Let's try to make an equation right away. The upstream time is one hour longer than the downstream time.

It is written like this:

Now, instead of each time, we substitute the formula:

We got the usual rational equation, let's solve it:

Obviously, the speed cannot be a negative number, so the answer is: km / h.

Relative motion

If some bodies are moving relative to each other, it is often useful to calculate their relative speed. It is equal to:

• the sum of the velocities, if the bodies are moving towards each other;
• the difference in velocities if the bodies are moving in the same direction.

Example # 1

Two cars drove out of points A and B at the same time towards each other at the speeds of km / h and km / h. In how many minutes they will meet. If the distance between points is km?

Solution I:

The relative speed of the vehicles is km / h. This means that if we are sitting in the first car, then it seems to us motionless, but the second car approaches us at a speed of km / h. Since the distance between the cars is initially km, the time after which the second car will pass the first:

Solution II:

The time from the start of the movement to the meeting of the cars is obviously the same. Let's designate it. Then the first car drove through the path, and the second -.

In total, they drove all the kilometers. Means,

Other traffic tasks

Example # 1:

A car drove from point A to point B. Simultaneously with him, another car drove out, which exactly half of the way was traveling at a speed that is km / h less than the first, and the second half of the way it traveled at a speed of km / h.

As a result, the cars arrived at point B at the same time.

Find the speed of the first car if it is known to be greater than km / h.

Solution # 1:

To the left of the equal sign, we write down the time of the first car, and to the right of the second:

Let's simplify the expression on the right side:

We divide each term by AB:

The result is the usual rational equation. Solving it, we get two roots:

Of these, only one is more.

Answer: km / h.

Example No. 2

A cyclist left point A of the circular track. In minutes he had not yet returned to point A and from point A a motorcyclist followed him. Minutes after departure, he caught up with the cyclist for the first time, and minutes after that he caught up with him a second time. Find the speed of the cyclist if the length of the track is km. Give your answer in km / h.

Solution:

Here we will equate distance.

Let the speed of the cyclist be, and the motorcyclist -. Until the moment of the first meeting, the cyclist was on the road for minutes, and the motorcyclist -.

At the same time, they drove equal distances:

Between the meetings, the cyclist traveled the distance, and the motorcyclist -. But at the same time, the motorcyclist drove exactly one more lap, this can be seen from the figure:

I hope you understand that they didn't actually go in a spiral - the spiral just schematically shows that they go in a circle, passing the same points of the track several times.

We solve the resulting equations in the system:

SUMMARY AND BASIC FORMULAS

1. Basic formula

2. Relative motion

• This is the sum of the velocities if the bodies are moving towards each other;
• the difference in velocities if the bodies are moving in the same direction.

3. Driving with the flow:

• If we move with the current, the speed of the current is added to our speed;
• if we move against the current, the current velocity is subtracted from the velocity.

We helped you figure out the traffic tasks ...

Now it's your turn ...

If you have carefully read the text and solved all the examples yourself, we are ready to argue that you understood everything.

And this is already half the way.

Write down in the comments if you figured out the tasks for the movement?

Which ones cause the greatest difficulties?

Do you understand that tasks for "work" are almost the same?

Write to us and good luck with your exams!

- Is it worth continuing the relationship if you and your partner have different speeds?

We are sitting in one of the small hotels in Nepal and, by tradition, we are playing a question. This is the last day in the mountains and the last time we pull up anonymous notes. We are 14 people from different countries and cities, we have just completed the trek to the Langtang Valley and to Lake Gosaykunda.

At the start, in Kathmandu, all the members of the track chipped in on an anonymous question. I - the presenter - took out one every evening and read the next problem aloud, which gave rise to a reason for discussion, and sometimes disputes - through the prism of different experiences, understanding the situation, or delusion - a matter of everyday life.

Our last evening in the mountains has come. Once again I unfold the piece of paper, read first to myself, and then to everyone:

"Is it worth continuing the relationship if you and your partner have different speeds?"

The sound of air being drawn into the lungs can already be heard. Over the three years of such conversations, the statistics have remained unchanged - questions about relationships have always been the most demanded. The group was preparing for a lively discussion.

But everyone was outstripped by that special quiet and calm timbre of voice, which occurs only in a person who does not need to prove anything:

- My thirty years of experience in marriage suggests that it is impossible to always have the same speed of movement with your partner, - said Olga, one of the participants in our hike. And she continued:

- One way or another, there will be moments when one will be faster and the other slower. And the situation will inevitably come when they switch places, of course, if we talk about long-distance relationships.

True, I have not heard anything - as well as other opinions, if they were at all that evening. Once every couple of years, if I'm lucky, life brings me to a phrase-book that infinitely unfolds its meaning. Once such a phrase became by accident somewhere seen: "One cannot find oneself, one can only create oneself." Words that not only stunned me to the depths of my soul, but literally turned my whole life upside down. That evening was special. I came across another phrase-book that could be read endlessly:

It is impossible to always have the same speed with your partner over a long distance.

I spent a long time then spinning around these words, trying to expand their meaning. I felt the truth behind them. But if with other phrases I had only to push myself off a little, as I was ready to write a whole book, then here it did not go further than a pleasant tickle, which is the point. I lacked the texture of my own experience. Then I came to Olga with a request to "beat off the serve." To answer my questions that arise around this topic.

Olga responded with ease.

About different speeds of movement of partners and relationships over a long distance

Submitted by Olesya Vlasova, author of the Re-Self blog. Married for 9 months (in a relationship - 3 years). Beats off - Olga Vakhrusheva, business consultant, married for 32 years. When we met, Olga was 15, and Nikolai was 18. They got married as soon as Olga turned 18. For 22 years they have been living in New Zealand, where they moved from Novosibirsk. Olga and Nikolai have two children and two grandchildren.

- What to do to the one who is faster? From the outside, the story that both partners cannot always have the same speed in a long-distance relationship sounds beautiful, and most importantly, one feels that these words are true, but from the inside everything is not so simple and obvious. What about the one who is ahead today? Should I help the second one? Or, on the contrary - to leave him alone and not "drag on himself"? And how to find peace of mind in such a situation?

- For me, the statement that in a long-distance relationship there cannot always be the same speed for both partners is an axiom. As well as the fact that two people building relationships are a priori different, two independent, unique personalities. Both are not perfect. But this is clear to me now.

When I was younger, I tried to build our intrafamily relations based on pre-unviable attitudes: we must always do everything together and in complete understanding, we must be one whole, love is a gift that happens to you, which you find if you're lucky ...

In practice, of course, everything turned out to be wrong. And attempts to tie reality to a far-fetched ideal caused misunderstandings, resentments, and quarrels, which could have been avoided if the original views of the world were more viable.

I do not know what is happening in young heads now and on what ideas your generation grew up, but in our time, girls from early childhood saw and heard something like the following:

• In fairy tales and in films: a prince on a white horse will surely gallop to the princess, he will love her more than life, they will always live happily, and he will solve all her problems.
• From the conversations of older women: a real man should ... And further down the list: earn, provide, be a support, be smart, caring, an excellent father, a loving husband, gentle, understanding, and so on. (in fact, many of these definitions are mutually exclusive).
• From the same source: real men were transferred to the world. You cannot count on them. Either drunkards, or lazy and henpecked, or heartless careerists. You need to keep everything under control and, in fact, you can trust a man with an eye.

So my head is a complete mess of performances. There is only hope that the perfect relationship will happen by itself or he will make you happy. But now it is clear that no one can make another person happy (no matter how hard he tries). This is an internal process that goes in parallel with steps towards each other.

Back to your main question. What should someone who is faster today do? The answer is I don’t know. There is no one-size-fits-all answer. Sometime you need to help, sometime leave alone, sometime set a guiding kick (with love). Often you just need to go about your business, not to panic, but make it clear that you are here, you are there and you worry and love. If we are talking about two adequate people, and not about pathology, then simply understanding that this is not forever usually helps a lot.

In addition, there are often objective reasons for a decrease in speed:

• The difference in temperaments (you have to learn to live with this if you want to keep the relationship).
• Health problems that a man often does not talk about, and a woman invents God knows what.
• Problems at work or in business (which he also usually does not talk about until he figured out what to do about it).
• Some big changes that need to be realized before taking the next step.
• The difference in age (and, accordingly, in speed).
• Hormonal changes.
• Fears, finally. Of which men have no less, and maybe more, than ours, but there is no one to go to for help.

And here we are with our own speed and personal growth. In general, as my experience shows, this question often arises among young girls.

- Let's talk about a young girl. She thinks (objectively or not, it's still a question), at least it seems to her that she is doing more - pulling work, children, home. But he is not. Does not help. Does less.

- Yes, it is familiar. It seems that he owes me. I earn money, and the children are on me. Claims. Expectations. After three years of life together begins - socks in the hallway, either said or did something.

We need to understand the reasons. Analyze. Is it a temporary decrease in speed or is it like lying on the couch? The second is unlikely to be close to an active girl in life. But the reasons may be different. Very often we ourselves do not give our men a chance to get involved in the process.

For example, we voiced the problem (and often did not voice it at all, but we hope that he will guess it himself). He has not yet had time to comprehend the problem, but we are already rushing to do and solve everything ourselves. Well, why should he then run with us in a race? Or - why then did you tell him about the problem?

Or he did something, and we are unhappy - he did something wrong. Well, once it’s not so, the second time it’s not so, and then you don’t want to move (would you want to?). Why not put the question differently: “This is my area of ​​responsibility, and this is yours. How and what you do is your decision, but the result is expected such and such. " He may stumble once, maybe he will forget, and then he will figure it out. If we believe that he will figure it out, and do not snort at every occasion.

This applies to everything. Starting from the elementary: instead of annoyed in his voice to declare that he never takes out the trash, and you do it yourself, yourself ... But you, too, get tired ... and further down the text. It is more productive to say: “Honey, do it like this: take out the trash in the house on you. I'm counting on you. " And that's all. And forget. And can't stand it. And not to remind. Even if the house starts to whine. He, too, will feel it, and remember, and throw it out, and will already remember.

It is also very important to set specific tasks for your partner and ask clearly and clearly what we need. In what we are waiting for help. They simply do not see many things. They don't even know about their existence at first. And our thoughts do not know how to read. It's much easier to say, "Honey, I'm stitching up in the kitchen, please hang up the laundry and put the kids to bed." If a man is adequate and is not busy at this moment with something important, then the issue is resolved. And what does a young woman usually do? He rushes between the kitchen, the laundry and the children, waiting for him to understand himself (this is obvious), becoming satanic, offended. And you could just say.

The same rules apply to your relationship with your son. Apparently, boys perceive such language better.

And it is important to realize such a simple thing that if at a given moment in a relationship a woman (or a man) is stronger, this does not mean that she (he) is always right (right).

- And about those who become weaker at some point and can reflect on it? After all, this is also difficult. A man by itself, but also a girl capable of introspection, will feel uneasy: for some reason she is not in a rut, maybe pregnancy, maybe, I don’t know, an illness or something, but he has a career, a rise, development, movement. This is jealousy, and anxiety, and just the feeling of one's own worthlessness can creep out. Have you had this?

- Yes, just when moving to New Zealand. From the very beginning, we relied on my husband. He had a language, and he immediately went to study and work. I came home tired, but on the rise and with a bunch of interesting information, acquaintances, plans. And I felt completely lost. I couldn't do the simplest things myself (I don't have a language, I don't drive a car, how the bank works, I don't know, I have no acquaintances, my husband cannot provide support - he is not at home all day, there are two small children in his arms). And a month ago I owned businesses, consulted people, taught, taught others what to do and how to do it.

The realization that this is happening to me helped. That is, it is important not to deceive yourself and not look for the guilty, but with maximum honesty describe the situation in which I am at the moment.

• What's happening? Where am I now?
• Is this a temporary inconvenience or a real problem?
• How did I get here?
• What does not suit me in the situation?
• What can I do to change the situation?
• Map out real steps.
• Take these steps.
• Check the result against the target, make corrections, if necessary.
• Move on.

In principle, I solve all my problems using this algorithm. The most difficult thing is usually to become aware of your emotions, to take yourself out of the situation emotionally and turn on your head. Sometimes I give myself permission for another week to “try and feel sorry for myself,” and then get down to business. Usually works.

Trying to ignore your emotions and fears certainly doesn't work. It's easier for me to say to myself: “Ok, I'm afraid of this scenario. Good. Hello fear. " Then ask yourself the question: “What will happen in the worst case if the fears are justified? Is it deadly? What would be option B? Can I live with this? " Most often, the answer is that you can live with it and not everything is scary in reality. And then the energy appears to look for options and move on.

The first months in New Zealand were painful to be completely zeroed out, the loss of social contacts, status, skills, understanding of how to earn money, how life and society work, the transformation from a sociable professional into a silent "nothing". But there were children in her arms, so it was impossible to go into complete hysterics. Therefore, after a month I went to learn the language (as - a separate detective story). Six months later, she went to work as a volunteer in a bureau for supporting poor families (she overcame the fear of communication, gained local experience, acquaintances), and six months later she went to work in her specialty. Well, go ahead.

- What is the most important thing in a long-distance relationship?

- From what I have seen in my life, from communication with couples who have lived a long life together and are happy together (and there are plenty of them, by the way, but this is somehow very little said in modern media, more and more about problems ), - a simple tendency is very clear in the relationship of these couples.

All happy couples have mutual trust. I have not seen a single couple so that people do not trust each other and live happily. It is impossible to live with a person and constantly expect a catch. This is a life of endless fear and stress. For both.

I also know couples where everything is not easy. Mistrust fills their world. From the outside, it can be seen that the most distrustful one usually has big problems with self-esteem, and besides, he (herself) is guilty of exactly what he suspects his half of, or had a very bad life experience, or the expectations are very unrealistic.

That is, we again return to the question of our own fears, unrealistic expectations and other cockroaches in our heads. The partner most often has nothing to do with it. You need to deal with yourself. In certain cases, you probably need to contact a specialist who can help specific people in a specific situation.

- How can one gain basic trust? Have you worked on this?

- I was lucky: I never lost it. The feeling of a shoulder and a covered back was fundamental for me from the very beginning of the relationship. And it was this that helped me to go through different stages, including the sections on which we moved at different speeds. I know that my man will never go for deep, thoughtful meanness, that he will act in accordance with his basic principles and his nature. So I perceive any problems and misunderstandings as problems and misunderstandings. If the base is trust and the absence of a knife in the back, then everything else can be solved. I guess I can say that my trust is a choice. And I do it every day.

- And jealousy?

- If, deep down, you understand that anything can happen in life, and you are ready to let your man go in a situation where his happiness will be somewhere else, then the reason for jealousy disappears.

In this regard, the question of lies in relationships arises. The more you strive to control each step of your partner, the more you dream of merging into a single whole and do not leave him personal space, the more he needs to lie and dodge. Sometimes - so as not to disturb you, sometimes - because it's easier, it happens because you don't understand how it should be. I know from myself as a child. I grew up with an extremely controlling mother, where the forces were unequal, and I am not one of those who follow the lead. So, if possible, save your loved one from the very need to lie, give him space, the opportunity not to answer all the questions you ask and not to report on every step. The more you believe in your man and in your man, the better and more comfortable you both are.

It is very important to learn to respect the decisions of your man. We do not always understand the logic, causes and expected consequences, but not everything needs to be understood intellectually. This is also a necessary component of trust, and this had to be learned.

- Olga, do you and your husband look alike? What is your conclusion after so many years together?

- No, we are not alike.

- So how to be with someone who is not like you? What to do with this dissimilarity?

- We are not alike, but we complement each other. I am very interested in his view of problems and situations. I'm just interested and warm with him. He is constantly generating ideas. He makes many things look from a different angle and from the other side. You begin to understand that there can be different answers to the same question, and they both have a right to exist. We can accept that we disagree on some issue. This approach makes living together very interesting and deprives them of any reason for conflict.

This dissimilarity can be enjoyed. Get high. Definitely not trying to avoid or smooth it out (tested - doesn't work). As with everything, the first step is to recognize where you are not alike. Does this complement and enrich your shared “we” or are these fundamental differences that you cannot be together with? If the differences are fundamental and you are incompatible, the answer is clear - the sooner a couple understands this, the better.

If these are just two different "I", then why not a task for personal growth? Learn to enjoy your differences, learn to be flexible, learn to be tolerant of the person you are closest to. Probably, next to the dissimilar, you can learn much more. See and get to know yourself from a completely different side.

- You started a relationship at a very early age. And these are colossal personal changes - the way you are at 18, at 28, or at 48. Completely different people, as a rule. How can we continue to love each other despite all these changes?

- While both of you are growing, changing, learning, talking about problems, overcoming them together, raising children, doing a joint work, reading and discussing, relaxing, you are developing a huge joint story, gratitude to each other for the hand outstretched in time, for the warmth, for a hint, for love, for faith ... I think that this joint growth only brings us closer. The main thing is that you talk to each other when something went wrong, and do not move in fundamentally opposite directions.

- I was preparing for the meeting and with horror stumbled upon the thought of my early youth that divorce is normal. Like, if something goes wrong - a divorce. This is fine. I don't know what it was. Or the consequences of an era when a new level of openness and accessibility created this trend. Or the lack of good examples before my eyes ... But I can remember myself as a 20-year-old, seriously talking about this. And it seems like it's really okay to disperse, if it really happened. But something else horrified me - along with thinking about divorces, there was not a single thought that, in fact, building relationships is much more normal. Working on them, strengthening, making a conscious contribution, the need to go through difficult sections. Have you instilled thoughts about such work in your children? And how important is it to talk about it?

- I think it is vital. It is important to teach children this, and even better - to show by example. That is, it is not enough to say, it is imperative to live your life as you say. Children feel false a mile away, and absorb emotions and family atmosphere like sponges. What was a torment and a search for Nikolai and me becomes obvious things for them.

My children and I talked and talk a lot about this, especially in adolescence and now, when they are building their relationships and raising their children. By the way, both say that at some point our example caused difficulties, since the bar was set too high. What is obvious and understandable to them is not obvious to their other half.

It would be great if moms and society more often voiced things like this:

• Happy, harmonious relationships don't "happen" - they are built by two loving people.
• Before entering into a long-term relationship, define your expectations. Try to understand what is important for you now and in future life (children - their absence, career - home, life in a big city - on an island in the ocean, gentle - grasping). It is clear that this will all change many times, but trying to understand your life priorities helps a lot.
• Check the coordinates with your chosen one. Do you agree on the most important issues?
• Your half is a living person, not an ideal. With all the ensuing consequences. In certain situations, you may not like him, and this is normal and does not mean the death of the relationship. It's like with children. I really love my children, but this does not mean that I always like them and in everything. (Do I understand it?)
• He may not always want what you want (and vice versa).
• Your half is not your copy, but the other person. Your task is to hear and understand it. Although it will most likely not be possible to fully understand. So take this difference as a fact of life and don't try to redo it (basic personality traits, I'm not talking about socks in the hallway).
• The state of happiness and harmony in a relationship is not constant. It comes and goes, but it certainly comes back if the couple does not scatter at the first problematic situation. And with each such return, feelings become deeper and more tender (we have gone through so much together, we have already understood so much about each other).

- Before the first quarrel, it seems that the relationship will always be smooth, small roughnesses do not count, after the first quarrel it seems that it will never go away and that this scar is forever. Both you and your partner. Comment from the height of your experience.

- To quarrel without offending is also a science, it will come with time, but there will also be breakdowns. We perceive the same words in different ways. One and the same thought can be presented in such a way as to seek a joint solution, or it can be done in such a way that both will lick the scars. The tone is important, the moment is important, how the phrase is constructed is important. You need to understand why the fight happened - because you are tired, sick, overheated, or is there a structural problem in the family that needs to be addressed? It is very important not to get personal. We women suffer from this often.

What can we do about it? How to avoid such passions in the future? How can we talk about a sick person without offending or blaming? Why did you (me) have such a reaction to the remark (question)? I didn’t put such a meaning into it, I didn’t mean it. There can be anything - childhood fears, previous negative experiences, wrong guesses and thinking out thoughts, our tone and construction of the question. We need to talk about this. Often not immediately, but when the fuse has cooled down and both of you have calmed down. But leaving such things unimaginable is dangerous.

On the other hand, it is desirable to learn how to treat everything easier. (Oh, how long did it get to me.) Not trying to be perfect, not trying to build perfect relationships, giving yourself and others the right to make mistakes. To understand that swearing and putting up is normal (the question is how it happens), that there will never be a complete understanding (this is a myth). Learn not to make an elephant out of a fly. Many "problems" do not need to be corrected or deeply reflected about them, it is better to simply forget (as they say, "we passed, and that's it").

In short, for all the seriousness of the issue, try not to take your life together and relationships too seriously. And you do not need to persistently and endlessly improve everything (yourself, him, relationships), often our imperfections are the highlight that keeps us together.

Woman: "Deliver your loved ones from your claims and expectations."

Man: “Don't forget that your husband is also human. Do not take his brains out unless absolutely necessary. "

Somehow like this.

For a snack, I want to voice an important thought for me, which does not directly relate to your questions and, perhaps, until it causes a resonance.

Sometime in real life, we all face death, come to the edge and realize (not with the mind, but with the heart) that we are all here temporarily. Both ourselves and the people we love. After such "insight" (if you do not hide your head in the sand from fear) comes a more careful attitude towards yourself and those who are nearby, and the ability to appreciate the banal little things in life, and most importantly - to receive joy and pleasure from them. It makes life beautiful and filled with love. Maybe if you filter your reactions, relationships, problems, fears through the filter of mortality, then many questions that seem serious will go away by themselves.

Hug tightly.

In addition to the topic, Olga prepared for an independent analysis in the field of relationships and a better understanding of both herself and her man.

Olesya Vlasova

P.S. Friends, for 5 years now we have been conducting retreats, expeditions and mountain tracks in different parts of Asia. The goal of our programs is to release the mind and body from tension, restore strength and launch the rhythm of conscious changes for the better. Our tools are yoga, meditation, freediving, the practice of silence, the right atmosphere for full-fledged switching and a kind company of like-minded people. If you were looking for a place where you can fully switch and qualitatively rethink the current "settings" - we are here.

In the previous problems on movement in one direction, the movement of bodies began simultaneously from the same point. Let us consider the solution of problems on movement in one direction, when the movement of bodies begins simultaneously, but from different points.

Let a cyclist and a pedestrian leave points A and B, the distance between which is 21 km, and go in the same direction: a pedestrian at a speed of 5 km per hour, a cyclist at 12 km per hour

12 km per hour 5 km per hour

A B

The distance between a cyclist and a pedestrian at the moment of their start is 21 km. For an hour of their joint movement in one direction, the distance between them will decrease by 12-5 = 7 (km). 7 km per hour - speed of convergence of a cyclist and a pedestrian:

A B

Knowing the speed of convergence of a cyclist and a pedestrian, it is not difficult to find out how many kilometers the distance between them will decrease after 2 hours, 3 hours of their movement in the same direction.

7 * 2 = 14 (km) - the distance between a cyclist and a pedestrian will decrease by 14 km after 2 hours;

7 * 3 = 21 (km) - the distance between the cyclist and the pedestrian will decrease by 21 km after 3 hours.

Every hour the distance between cyclist and pedestrian decreases. After 3 hours, the distance between them becomes equal to 21-21 = 0, i.e. the cyclist will catch up with the pedestrian:

A B

In catch-up problems, we deal with the quantities:

1) the distance between the points from which the simultaneous movement begins;

2) the speed of convergence

3) the time from the moment of the beginning of the movement until the moment when one of the moving bodies overtakes the other.

Knowing the meaning of two of these three quantities, one can find the meaning of the third quantity.

The table contains the conditions and solutions of problems that can be compiled to “catch up” by a cyclist a pedestrian:

	The speed of convergence of a cyclist and a pedestrian in km per hour	Time from the moment of starting the movement until the moment when the cyclist overtakes the pedestrian, in hours	Distance from A to B in km

Let us express the relationship between these values ​​by the formula. Let us denote by the distance between the points and, - the speed of approach, the time from the moment of exit to the moment when one body overtakes another.

In “catch-up” problems, the approach speed is most often not given, but it can be easily found from the data of the problem.

Task. The cyclist and the pedestrian left simultaneously in the same direction from two collective farms, the distance between which is 24 km. A cyclist was traveling at a speed of 11 km per hour, and a pedestrian was walking at a speed of 5 km per hour. How many hours after leaving the cyclist will overtake the pedestrian?

To find how long after his exit the cyclist will catch up with the pedestrian, you need to divide the distance that was between them at the beginning of the movement by the speed of approach; the speed of approach is equal to the difference between the speeds of the cyclist and the pedestrian.

Solution formula: = 24: (11-5); = 4.

Answer. After 4 hours the cyclist will overtake the pedestrian. Conditions and solutions of inverse problems are written in the table:

	Cyclist speed in km per hour	Pedestrian speed in km per hour	Distance between collective farms in km	Time per hour

Each of these tasks can be solved in other ways, but they will be irrational in comparison with these solutions.

Page 1

Starting in the 5th grade, students often encounter these problems. Even in elementary school, students are given the concept of "general speed". As a result, they form not entirely correct ideas about the speed of convergence and the speed of removal (this terminology is not in elementary school). Most often, solving a problem, students find the amount. It is best to start solving these problems with the introduction of the concepts: "convergence rate", "removal rate". For clarity, you can use the movement of the hands, explaining that bodies can move in one direction and in different directions. In both cases, there can be both the speed of approach and the speed of removal, but in different cases they are found in different ways. After that, students write down the following table:

Table 1.

Methods for finding the speed of convergence and the speed of removal

	Moving in one direction	Movement in different directions
Removal rate
Approach speed

When analyzing the problem, the following questions are given.

Using the movement of the hands, we find out how the bodies move relative to each other (in one direction, in different directions).

We find out what action is the speed (addition, subtraction)

Determine what speed it is (approach, removal). We write down the solution to the problem.

Example # 1. From cities A and B, the distance between which is 600 km, at the same time, trucks and cars came out towards each other. The speed of the car is 100 km / h, and the speed of the cargo is 50 km / h. In how many hours will they meet?

Students use hand movements to show how cars move and draw the following conclusions:

cars move in different directions;

the speed will be added;

since they are moving towards each other, this is the speed of convergence.

100 + 50 = 150 (km / h) - approach speed.

600: 150 = 4 (h) - travel time before the meeting.

Answer: in 4 hours

Example # 2. The man and the boy left the state farm for the vegetable garden at the same time and walk the same road. The speed of the man is 5 km / h, and the speed of the boy is 3 km / h. What is the distance between them in 3 hours?

With the help of hand movements, we find out:

a boy and a man are moving in the same direction;

the speed is found by the difference;

the man walks faster, that is, moves away from the boy (removal rate).

Relevant about education:

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2. BODY SPEED. STRAIGHT AND UNIFORM MOTION.

Speed Is a quantitative characteristic of body movement.

average speed Is a physical quantity equal to the ratio of the point displacement vector to the time interval Δt during which this displacement occurred. The direction of the average velocity vector coincides with the direction of the displacement vector. Average speed is determined by the formula:

Instant speed, that is, the speed at a given moment in time is a physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval Δt:

In other words, the instantaneous speed at a given moment in time is the ratio of a very small movement to a very small period of time during which this movement occurred.

The instantaneous velocity vector is directed tangentially to the trajectory of the body's motion (Fig. 1.6).

Rice. 1.6. Instantaneous velocity vector.

In the SI system, speed is measured in meters per second, that is, the unit of speed is considered to be the speed of such a uniform rectilinear motion, in which in one second the body travels a path of one meter. The speed unit is denoted m / s... Velocity is often measured in other units. For example, when measuring the speed of a car, train, etc. the commonly used unit is kilometer per hour:

1 km / h = 1000 m / 3600 s = 1 m / 3.6 s

1 m / s = 3600 km / 1000 h = 3.6 km / h

Speed ​​addition (perhaps not necessarily the same question will be in 5).

The velocities of body movement in different reference frames are linked by the classical speed addition law.

Body speed relative fixed frame of reference is equal to the sum of the body's velocities in moving frame of reference and the most mobile frame of reference relative to the stationary one.

For example, a passenger train travels on a railroad at a speed of 60 km / h. A person walks along the carriage of this train at a speed of 5 km / h. If we consider the railway stationary and take it as a reference system, then the speed of a person relative to the reference system (that is, relative to the railway) will be equal to the addition of the speeds of the train and the person, that is

60 + 5 = 65 if the person goes in the same direction as the train

60 - 5 = 55 if the person and the train are moving in different directions

However, this is only true if the person and the train are moving along the same line. If a person moves at an angle, then this angle will have to be taken into account, remembering that speed is vector quantity.

The example is highlighted in red + The law of addition of displacement (I think this does not need to be learned, but for general development you can read it)

Now let's look at the example described above in more detail - with details and pictures.

So, in our case, the railway is fixed frame of reference... The train that moves this road is moving frame of reference... The carriage on which the person is walking is part of the train.

The speed of a person relative to the car (relative to the moving frame of reference) is 5 km / h. Let us denote it by the letter Ch.

The speed of the train (and hence the carriage) relative to the stationary frame of reference (that is, relative to the railroad) is 60 km / h. Let us denote it by the letter B. In other words, the speed of the train is the speed of the moving frame of reference relative to the stationary frame of reference.

The speed of a person relative to the railway (relative to a stationary frame of reference) is still unknown to us. Let's designate it with a letter.

Let's connect the XOY coordinate system with the stationary reference system (Fig. 1.7), and the X P O P Y P. coordinate system with the moving reference system. Now let's try to find the speed of a person relative to the stationary reference system, that is, relative to the railroad.

For a short time interval Δt, the following events occur:

Then, during this period of time, the movement of a person relative to the railway:

it addition law of displacements... In our example, the movement of a person relative to the railroad is equal to the sum of the movements of a person relative to the carriage and the carriage relative to the railroad.

Rice. 1.7. The law of addition of displacements.

The law of addition of displacements can be written as follows:

= Δ H Δt + Δ B Δt

The speed of a person relative to the railroad is:

Person's speed relative to the car:

Δ H = H / Δt

Car speed relative to the railroad:

Therefore, the speed of a person relative to the railway will be equal to:

This is the lawspeed addition:

Uniform movement- this is movement with constant speed, that is, when the speed does not change (v = const) and acceleration or deceleration does not occur (a = 0).

Straight motion- this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.

Uniform rectilinear movement- This is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.

The speed of uniform rectilinear movement does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:

Uniform straight motion speed Is a physical vector quantity equal to the ratio of the body's displacement over any time interval to the value of this interval t:

Thus, the speed of uniform rectilinear motion shows how much a material point moves per unit of time.

Moving with uniform rectilinear motion is determined by the formula:

Distance traveled in rectilinear motion it is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of motion, then the projection of the velocity onto the OX axis is equal to the magnitude of the velocity and is positive:

v x = v, that is, v> 0

The projection of displacement on the OX axis is equal to:

s = vt = x - x 0

where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)

Equation of motion, that is, the dependence of the coordinates of the body on time x = x (t) takes the form:

If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body's velocity onto the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид.