How to find your own current speed. Water movement problems

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

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

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

Fourth task

Kolya goes to work by car at a speed of km/h. Colleague Kolya Vova is driving at a speed of km/h. Kolya lives kilometers away from Vova.

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

Did you count? Let's compare the answers - it turned out that Vova will catch up with Kolya in an hour or in minutes.

Let's compare our solutions...

The drawing looks like this:

Similar to yours? Well done!

Since the problem asks how long after the guys met, and they left at the same time, the time they drove will be the same, as well as the meeting place (in the figure it is indicated by a dot). When composing the 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 let's look at the axis of movement.

Let's start with the path that Kolya took. Its path () is shown in the figure as a segment. 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 took.

Based on these conclusions, we obtain the equation:

Got it? 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 from this example you understand how important the role is played Well done drawing!

And we smoothly move on, or rather, we have already moved on to the next point of our algorithm - bringing all quantities to the same dimension.

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

Dimension.

Problems do not always give the same dimension for each participant in the movement (as was the case in our easy problems).

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

We can’t just take and substitute the values ​​into the formula - the answer will be incorrect. Even in terms of units of measurement, our answer “fails” the reasonableness test. Compare:

Do you see? When multiplying correctly, we also reduce the units of measurement, and, accordingly, we obtain a reasonable and correct result.

What happens if we don’t convert to one measurement system? The answer has a strange dimension and the result is % incorrect.

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

Length units:

centimeter = millimeters

decimeter = centimeters = millimeters

meter = decimeters = centimeters = millimeters

kilometer = meters

Time units:

minute = seconds

hour = minutes = seconds

day = hours = minutes = seconds

Advice: When converting units of measurement related to time (minutes into hours, hours into seconds, etc.), imagine a clock dial 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. an hour, and a minute is an hour.

And now a very simple task:

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

Did you count? The correct answer is km.

minutes is an hour, and another minutes from an hour (mentally imagined a clock dial, and said that minutes is a quarter of an hour), respectively - min = hours.

Reasonableness.

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 can’t be negative, right? So, rationality, that’s what it’s about)

Calculation.

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

“Love of tables” or “when drawing is not enough”

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

First task

A cyclist and a motorcyclist left at the same time from point to point, the distance between them being kilometers. 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.

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

In general, draw, and now we’ll compare what you got.

It's a bit empty, isn't it? Let's draw a table.

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

True, we will add one more column - Name, about whom we write information - a motorcyclist and a cyclist.

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

Did you get a table like this?

Now let's analyze everything we have and at the same time enter the data into the table and figure.

The first thing we have is the path that the cyclist and motorcyclist took. It is the same and equal to km. Let's bring it in!

Let's take the speed of the cyclist as, then the speed of the motorcyclist will be...

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

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

That's right, divide the distance by the speed. Enter this into the table.

Now our table is filled in, now we can enter the data into the drawing.

What can we reflect on it?

Well done. Speed ​​of movement of motorcyclist and cyclist.

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

What data is not reflected in the table or figure?

Right. The time the motorcyclist arrived before the cyclist. We know that the time difference is minutes.

What should we do next? That's right, convert 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 - manipulations leading to the only correct answer.

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

Drawing up the equation:

Look at your table, at the last condition that is not included in it and think, the relationship between what and what can we put into the equation?

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

Logical? The cyclist rode more; if we subtract the motorcyclist’s time from his time, we will get the difference given to us.

This equation is rational. If you don’t know what this is, read the topic “”.

We bring the terms to a common denominator:

Let's open the brackets and present similar terms: Phew! Got it? Try your hand at the following problem.

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. Let's decide!

We received two possible answers. Let's see what we got for? That's right, the speed of the cyclist.

Let us remember the “3P” rule, more specifically “reasonableness”. Do you know what I mean? Exactly! Speed ​​cannot be negative, so our answer is km/h.

Second task

Two cyclists set out on a -kilometer ride at the same time. The first one drove at a speed that was one km/h faster than the second one, and arrived at the finish line hours earlier than the second one. Find the speed of the cyclist who came second to the finish line. Give your answer in km/h.

Let me remind you of the solution algorithm:

• Read the problem a couple of times and understand all the details. Got it?
• Start drawing a picture - in which direction are they moving? how far did they travel? Did you draw it?
• Check that all your quantities are of the same dimension and begin to briefly write out the conditions of the problem, making 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 up an equation and solve. Yes, and finally - remember the “3Rs”!
• I've done everything? Well done! I found out that the speed of the cyclist is km/h.

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

Let's continue our conversation. So what is the speed of the first cyclist? km/h? I really hope you’re not nodding yes now!

Read the question carefully: “What is the speed of first cyclist?

Do you understand what I mean?

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

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

One more 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 dimensions not only during the solution itself, but also when writing down the answers.

Circular movement problems

Bodies in problems can move not necessarily straight, but also in a circle, for example, cyclists can ride along a circular track. Let's look at this problem.

Task No. 1

A cyclist left a point on the circular route. Minutes later, he had not yet returned to the point and the motorcyclist left the point after him. Minutes after leaving, he caught up with the cyclist for the first time, and minutes after that he caught up with him for the second time.

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

Solution to problem No. 1

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

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

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

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

Got it? Try to solve the following problems yourself:

Tasks for independent work:

1. Two motorcycles start at the same time in one direction of the two dia-metral-but-pro-ti-on- false points of a circular route, the length of which is equal to km. After how many minutes do the cycles become equal for the first time, if the speed of one of them is km/h higher than the speed of the other? ho-ho?
2. From one point on a circular highway, the length of which is equal to km, at one time there are two motorcyclists in the same direction. The speed of the first motorcycle is equal to km/h, and minutes after the start it was ahead of the second motorcycle by one lap. Find the speed of the second motorcycle. Give your answer in km/h.

Solutions to problems for independent work:

1. Let km/h be the speed of the first motor cycle, then the speed of the second motor cycle is equal to km/h. Let the cycles be equal for the first time in a few hours. In order for the cycles to be equal, the faster one must overcome them from the beginning distance equal to the length of the route.

   We get that the time is 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, so we get the equation:

   The speed of the second motorcyclist is km/h.

Current problems

Now that you are excellent at solving problems “on land,” let’s move into the water and look at the scary problems associated with the current.

Imagine that you have a raft and you lower it into the lake. What's happening to him? Right. It stands because a lake, a pond, a puddle, after all, is still water.

The current speed in the lake is .

The raft will only move if you start rowing yourself. The speed it acquires will be the raft's own speed. It doesn’t matter where you swim - left, right, the raft will move at the speed with which you row. It's clear? It's logical.

Now imagine that you are lowering a raft onto the river, you turn away to take the rope..., you turn around, and it... floats away...

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

Its speed is zero (you are standing in shock on the shore and not rowing) - it moves at the speed of the current.

Got it?

Then answer this question: “At what speed will the raft float down the river if you sit and row?” Thinking about it?

There are two possible options here.

Option 1 - you go with the flow.

And then you swim at your own speed + the speed of the current. The flow seems to help you move forward.

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

Hard? That's right, because the current is trying to “throw” you back. You are making more and more efforts to swim at least meters, respectively, the speed at which you move is equal to your own speed - the speed of the current.

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

Let's solve the problem and check.

Let's add to our path data on the speed of the current - km/h and the raft's own speed - km/h. How much time will you spend moving with and against the current?

Of course, you coped with this task without difficulty! It takes an hour with the current, and an hour against the current!

This is the whole essence of the tasks at movement with the current.

Let's complicate the task a little.

Task No. 1

A boat with a motor took an hour to travel from point to point, and an hour to return.

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

Solution to problem No. 1

Let us 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 takes the same path, respectively:

What did we charge for?

Current speed. Then this will be the answer :)

The speed of the current is km/h.

Task No. 2

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

Determine (in km/h) the kayak's own speed if it is known that the speed of the river is km/h.

Solution to problem No. 2

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

Are all quantities expressed in the same form? No. Our rest time is indicated in both hours and minutes.

Let's convert this into hours:

hour minutes = h.

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

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

Let's write down this data, as well as the path (as you understand, it is the same) and time, expressed in terms of 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 spent on its journey:

Did she swim for all the hours? Let's reread the task.

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

h the kayak really floated.

Let's bring all the terms to a common denominator:

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

I think you can handle this on your own too. What answer did you get? I have km/h.

Let's sum it up

ADVANCED LEVEL

Movement tasks. Examples

Let's consider examples with solutionsfor each type of task.

Moving with the Current

Some of the simplest tasks are river navigation problems. Their whole essence is as follows:

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

Example #1:

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

Solution #1:

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

We will enter all the 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 of the table. 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, we use immediately formula for distance:

Often you have to use formula for time:

Example #2:

A boat travels a distance of kilometers against the current an hour longer than with the current. Find the speed of the boat in still water if the speed of the current is km/h.

Solution #2:

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

It is written like this:

Now, instead of each time, let’s substitute the formula:

We have received an ordinary rational equation, let’s solve it:

Obviously, 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 velocities if bodies move towards each other;
• speed differences if bodies move in the same direction.

Example No. 1

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

I solution method:

Relative speed of cars km/h. This means that if we are sitting in the first car, it seems motionless to us, but the second car is approaching us at a speed of km/h. Since the distance between the cars is initially km, the time it will take for the second car to pass the first:

Method II:

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

In total they covered all the kilometers. Means,

Other movement tasks

Example #1:

A car left point A to point B. At the same time, another car left with him, which drove exactly half the way at a speed of km/h less than the first, and drove the second half of the way 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 that it is 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:

Let's divide each term by AB:

The result is an ordinary rational equation. Having solved it, we get two roots:

Of these, only one is larger.

Answer: km/h.

Example No. 2

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

Solution:

Here we will equate the distance.

Let the speed of the cyclist be, and the speed of 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 traveled equal distances:

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

I hope you understand that they didn’t actually drive in a spiral; the spiral just schematically shows that they drive in a circle, passing the same points on the route 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 speeds if the bodies move towards each other;
• difference in speed if bodies move in the same direction.

3. Moving 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 speed of the current is subtracted from the speed.

We helped you deal with movement problems...

Now it's your turn...

If you carefully read the text and solved all the examples yourself, we are willing to bet that you understood everything.

And this is already half the way.

Write below in the comments, have you figured out the movement problems?

Which ones cause the most difficulties?

Do you understand that tasks for “work” are almost the same thing?

Write to us and good luck on your exams!

This material is a system of tasks on the topic “Movement”.

Goal: to help students more fully master the technology of solving problems on this topic.

Problems involving movement on water.

Very often a person has to move on water: a river, lake, sea.

At first he did it himself, then rafts, boats, and sailing ships appeared. With the development of technology, steamships, motor ships, and nuclear powered ships came to the aid of man. And he was always interested in the length of the path and the time spent on overcoming it.

Let's imagine that it's spring outside. The sun melted the snow. Puddles appeared and streams ran. Let's make two paper boats and launch one of them into a puddle, and the second into a stream. What will happen to each of the boats?

In a puddle the boat will stand still, but in a stream it will float, since the water in it “runs” to a lower place and carries it with it. The same thing will happen with a raft or boat.

In a lake they will stand still, but in a river they will float.

Let's consider the first option: a puddle and a lake. The water in them does not move and is called standing.

The ship will float across the puddle only if we push it or if the wind blows. And the boat will begin to move in the lake with the help of oars or if it is equipped with a motor, that is, due to its speed. This movement is called movement in still water.

Is it different from driving on the road? Answer: no. This means that you and I know how to act in this case.

Problem 1. The speed of the boat on the lake is 16 km/h.

How far will the boat travel in 3 hours?

Answer: 48 km.

It should be remembered that the speed of a boat in still water is called own speed.

Problem 2. A motor boat sailed 60 km across a lake in 4 hours.

Find the motorboat's own speed.

Answer: 15 km/h.

Problem 3. How long will it take a boat whose own speed

equal to 28 km/h to swim 84 km across the lake?

Answer: 3 hours.

So, To find the length of the path traveled, you need to multiply the speed by the time.

To find the speed, you need to divide the path length by the time.

To find the time, you need to divide the length of the path by the speed.

How is driving on a lake different from driving on a river?

Let's remember the paper boat in the stream. He swam because the water in him moved.

This movement is called going with the flow. And in the opposite direction - moving against the current.

So, the water in the river moves, which means it has its own speed. And they call her river flow speed. (How to measure it?)

Problem 4. The speed of the river is 2 km/h. How many kilometers does the river carry?

any object (wood chips, raft, boat) in 1 hour, in 4 hours?

Answer: 2 km/h, 8 km/h.

Each of you has swam in the river and remembers that it is much easier to swim with the current than against the current. Why? Because the river “helps” you to swim in one direction, and “gets in the way” in the other.

Those who cannot swim can imagine a situation when a strong wind blows. Let's consider two cases:

1) the wind is blowing at your back,

2) the wind blows in your face.

In both cases it is difficult to go. The wind at our back makes us run, which means our speed increases. The wind in our faces knocks us down and slows us down. The speed decreases.

Let's focus on moving along the river. We have already talked about a paper boat in a spring stream. The water will carry it along with it. And the boat, launched into the water, will float at the speed of the current. But if she has her own speed, then she will swim even faster.

Therefore, to find the speed of movement along the river, it is necessary to add the boat’s own speed and the speed of the current.

Problem 5. The boat's own speed is 21 km/h, and the speed of the river is 4 km/h. Find the speed of the boat along the river.

Answer: 25km/h.

Now imagine that the boat must sail against the current of the river. Without a motor or even oars, the current will carry her in the opposite direction. But, if you give the boat its own speed (start the engine or seat the rower), the current will continue to push it back and prevent it from moving forward at its own speed.

That's why To find the speed of the boat against the current, it is necessary to subtract the speed of the current from its own speed.

Problem 6. The speed of the river is 3 km/h, and the boat’s own speed is 17 km/h.

Find the speed of the boat against the current.

Answer: 14 km/h.

Problem 7. The ship's own speed is 47.2 km/h, and the speed of the river is 4.7 km/h. Find the speed of the ship downstream and against the current.

Answer: 51.9 km/h; 42.5 km/h.

Problem 8. The speed of a motor boat downstream is 12.4 km/h. Find the boat's own speed if the speed of the river is 2.8 km/h.

Answer: 9.6 km/h.

Problem 9. The speed of the boat against the current is 10.6 km/h. Find the boat's own speed and the speed along the current if the speed of the river is 2.7 km/h.

Answer: 13.3 km/h; 16 km/h.

The relationship between speed with the current and speed against the current.

Let us introduce the following notation:

V s. - own speed,

V current - flow speed,

V according to flow - speed with the current,

V flow flow - speed against the current.

Then we can write the following formulas:

V no current = V c + V current;

Vnp. flow = V c - V flow;

Let's try to depict this graphically:

Conclusion: the difference in speed along the current and against the current is equal to twice the speed of the current.

Vno current - Vnp. flow = 2 Vflow.

Vflow = (Vflow - Vnp.flow): 2

1) The speed of the boat against the current is 23 km/h, and the speed of the current is 4 km/h.

Find the speed of the boat along the current.

Answer: 31 km/h.

2) The speed of a motor boat along the river is 14 km/h, and the speed of the current is 3 km/h. Find the speed of the boat against the current

Answer: 8 km/h.

Task 10. Determine the speeds and fill out the table:

* - when solving item 6, see Fig. 2.

Answer: 1) 15 and 9; 2) 2 and 21; 3) 4 and 28; 4) 13 and 9; 5)23 and 28; 6) 38 and 4.

Solving problems involving “moving on water” is difficult for many. There are several types of speeds, so the decisive ones are starting to get confused. To learn how to solve problems of this type, you need to know definitions and formulas. The ability to draw diagrams greatly facilitates the understanding of the problem and contributes to the correct composition of the equation. And a correctly composed equation is the most important thing in solving any type of problem.

Instructions

In the tasks of “moving along a river” there are speeds: own speed (Vc), speed with the current (Von flow), speed against the current (Vstream flow), current speed (Vflow). It should be noted that a boat's own speed is its speed in still water. To find the speed along the current, you need to add your own speed to the current speed. In order to find the speed against the current, you need to subtract the speed of the current from your own speed.

The first thing you need to learn and know by heart is formulas. Write down and remember:

Vflow=Vс+Vflow.

Vpr. current = Vc-Vcurrent

Vpr. flow=Vflow. - 2Vcurrent

Vflow = Vpr. flow+2Vflow

Vflow = (Vflow - Vflow)/2

Vс=(Vflow+Vflowflow)/2 or Vс=Vflow+Vflow.

Using an example, we will look at how to find your own speed and solve problems of this type.

Example 1. The speed of the boat downstream is 21.8 km/h, and against the current is 17.2 km/h. Find the boat's own speed and the speed of the river.

Solution: According to the formulas: Vс = (Vflow + Vflow flow)/2 and Vflow = (Vflow - Vflow flow)/2, we find:

Vtech = (21.8 - 17.2)/2=4.62=2.3 (km/h)

Vс = Vpr current+Vcurrent=17.2+2.3=19.5 (km/h)

Answer: Vc=19.5 (km/h), Vtech=2.3 (km/h).

Example 2. The steamer traveled 24 km against the current and returned, spending 20 minutes less on the return journey than when moving against the current. Find its own speed in still water if the current speed is 3 km/h.

Let's take the ship's own speed as X. Let's create a table where we will enter all the data.

Against the flow With the flow

Distance 24 24

Speed ​​X-3 X+3

time 24/ (X-3) 24/ (X+3)

Knowing that the steamer spent 20 minutes less time on the return journey than on the journey downstream, we will compose and solve the equation.

20 min = 1/3 hour.

24/ (X-3) – 24/ (X+3) = 1/3

24*3(X+3) – (24*3(X-3)) – ((X-3)(X+3))=0

72Х+216-72Х+216-Х2+9=0

X=21(km/h) – the ship’s own speed.

Answer: 21 km/h.

note

The speed of the raft is considered equal to the speed of the reservoir.

Attention, TODAY only!

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According to the mathematics curriculum, children are required to learn how to solve motion problems while still in primary school. However, problems of this type often cause difficulties for students. It is important that the child understands what his own speed , speed currents, speed downstream and speed against the tide. Only under this condition will the student be able to easily solve motion problems.

You will need

• Calculator, pen

Instructions

1. Own speed- This speed boats or other means of transportation in static water. Label it – V proper. The water in the river is in motion. So she has her own speed, which is called speed yu current (V current) Designate the speed of the boat along the river current - V along the current, and speed against the current – ​​V ave. flow.

2. Now remember the formulas needed to solve motion problems: V ex. flow = V proper. – V flow. V flow = V own. + V current

3. It turns out, based on these formulas, the following conclusions can be made. If the boat moves against the flow of the river, then V proper. = V flow current + V current. If the boat moves with the current, then V own. = V according to flow – V current

4. Let's solve several problems on moving along a river. Problem 1. The speed of the boat against the river current is 12.1 km/h. Discover your own speed boats, knowing that speed river flow 2 km/h. Solution: 12.1 + 2 = 14. 1 (km/h) – own speed boats. Task 2. The speed of the boat along the river is 16.3 km/h, speed river flow 1.9 km/h. How many meters would this boat travel in 1 minute if it was in still water? Solution: 16.3 – 1.9 = 14.4 (km/h) – own speed boats. Let's convert km/h to m/min: 14.4 / 0.06 = 240 (m/min). This means that in 1 minute the boat would travel 240 m. Problem 3. Two boats set off at the same time opposite each other from 2 points. The 1st boat moved with the flow of the river, and the 2nd – against the current. They met three hours later. During this time, the 1st boat covered 42 km, and the 2nd – 39 km. Discover your own speed any boat, if it is known that speed river flow 2 km/h. Solution: 1) 42 / 3 = 14 (km/h) – speed movement along the river of the first boat. 2) 39 / 3 = 13 (km/h) – speed movement against the flow of the river of the second boat. 3) 14 – 2 = 12 (km/h) – own speed first boat. 4) 13 + 2 = 15 (km/h) – own speed second boat.

Movement tasks seem difficult only at first glance. In order to discover, say, speed ship's movements contrary to currents, it is enough to imagine the situation expressed in the problem. Take your child on a short trip along the river, and the student will learn to “click problems like nuts.”

You will need

• Calculator, pen.

Instructions

1. According to the current encyclopedia (dic.academic.ru), speed is a collation of the translational motion of a point (body), numerically equal, in the case of uniform motion, to the ratio of the distance traveled S to the intermediate time t, i.e. V = S/t.

2. In order to detect the speed of movement of a ship against the current, you need to know the ship's own speed and the speed of the current. Own speed is the speed of the ship in still water, say, in a lake. Let's denote it - V proper. The speed of the current is determined by the distance to which the river carries an object per unit of time. Let's denote it – V current.

3. In order to determine the speed of the vessel's movement against the current (V current flow), it is necessary to subtract the current speed from the vessel's own speed. It turns out that we have the formula: V flow current = V own. – V current

4. Let's find the speed of the ship's movement contrary to the flow of the river, if it is known that the ship's own speed is 15.4 km/h, and the speed of the river flow is 3.2 km/h. 15.4 - 3.2 = 12.2 (km/h ) – the speed of the vessel against the river flow.

5. In motion problems, it is often necessary to convert km/h to m/s. In order to do this, you need to remember that 1 km = 1000 m, 1 hour = 3600 s. Consequently, x km/h = x * 1000 m / 3600 s = x / 3.6 m/s. It turns out that in order to convert km/h to m/s you need to divide by 3.6. Say, 72 km/h = 72:3.6 = 20 m/s. To convert m/s to km/h you need to multiply by 3, 6. Let's say 30 m/s = 30 * 3.6 = 108 km/h.

6. Let's convert x km/h to m/min. To do this, remember that 1 km = 1000 m, 1 hour = 60 minutes. So x km/h = 1000 m / 60 min. = x / 0.06 m/min. Consequently, in order to convert km/h to m/min. must be divided by 0.06. Say, 12 km/h = 200 m/min. To convert m/min. in km/h you need to multiply by 0.06. Let's say 250 m/min. = 15 km/h

Helpful advice
Don't forget what units you use to measure speed.

Note!
Don't forget about the units in which you measure speed. To convert km/h to m/s, you need to divide by 3.6. To convert m/s to km/h, you need to multiply by 3.6. To convert km/h to m/min. must be divided by 0.06. To convert m/min. in km/h must be multiplied by 0.06.

Helpful advice
A drawing helps solve a movement problem.

According to the mathematics curriculum, children should learn to solve motion problems in elementary school. However, problems of this type often cause difficulties for students. It is important that the child understands what his own speed, speed currents, speed downstream and speed against the stream. Only under this condition will the student be able to easily solve movement problems.

You will need

• Calculator, pen

Instructions

Own speed- This speed boat or other vehicle in still water. Label it - V proper.
The water in the river is in motion. So she has her own speed, which is called speed yu current (V current)
Designate the speed of the boat along the river flow as V along the current, and speed against the current - V ave. flow.

Now remember the formulas necessary to solve motion problems:
V av. flow = V own. - V current
V according to flow = V own. + V current

So, based on these formulas, we can draw the following conclusions.
If the boat moves against the flow of the river, then V proper. = V flow current + V current
If the boat moves with the current, then V proper. = V according to flow - V current

Let's solve several problems about moving along a river.
Problem 1. The speed of the boat against the river current is 12.1 km/h. Find your own speed boats, knowing that speed river flow 2 km/h.
Solution: 12.1 + 2 = 14, 1 (km/h) - own speed boats.
Problem 2. The speed of the boat along the river is 16.3 km/h, speed river flow 1.9 km/h. How many meters would this boat travel in 1 minute if it was in still water?
Solution: 16.3 - 1.9 = 14.4 (km/h) - own speed boats. Let's convert km/h to m/min: 14.4 / 0.06 = 240 (m/min). This means that in 1 minute the boat would travel 240 m.
Problem 3. Two boats set off simultaneously towards each other from two points. The first boat moved with the flow of the river, and the second - against the flow. They met three hours later. During this time, the first boat traveled 42 km, and the second - 39 km. Find your own speed each boat, if it is known that speed river flow 2 km/h.
Solution: 1) 42 / 3 = 14 (km/h) - speed movement along the river of the first boat.
2) 39 / 3 = 13 (km/h) - speed movement against the river flow of the second boat.
3) 14 - 2 = 12 (km/h) - own speed first boat.
4) 13 + 2 = 15 (km/h) - own speed second boat.