How to find speed in still water. Tasks for movement to prepare for the exam in mathematics (2020). Solving problems for independent work

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

You will need

• Calculator, pen

Instruction

Own speed- This speed boat or other vehicle in still water. Designate it - V own.
The water in the river is in motion. So she has her speed, which is called speed th current (V current)
Designate the speed of the boat along the river - V along the current, and speed against the current - V pr. tech.

Now memorize the formulas needed to solve motion problems:
V pr. tech. = V own. - V tech.
V by current = V own. + V tech.

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

Let's solve several problems on the movement along the river.
Task 1. The speed of the boat against the current of the river is 12.1 km/h. Find your own speed boats, knowing that speed river current 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 current 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 \u003d 14.4 (km / h) - own speed boats. 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.
Task 3. Two boats set off simultaneously towards each other from two points. The first boat moved along the river, and the second - against the current. They met three hours later. During this time, the first boat covered 42 km, and the second - 39 km.Find your own speed each boat, if it is known that speed river current 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 current 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.

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

Why is it so? 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.

The fourth task

Kolya goes to work by car at a speed of km/h. Colleague Kolya Vova travels at a speed of km/h. Kolya lives at a distance of km from Vova.

How long will it take Vova to overtake 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 hours or minutes.

Let's compare our solutions...

The drawing looks like this:

Similar to yours? Well done!

Since the problem asks how long the guys met and 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). Making equations, take the time for.

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

Let's start with the path that Kolya did. 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 did.

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 that in this example you understand how important the role of well crafted drawing!

And we are smoothly moving on, or rather, we have already moved on to the next step in our algorithm - bringing all quantities to the same dimension.

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

Dimension.

Not always in tasks the same dimension is given for each participant in the movement (as it was in our easy tasks).

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

We can't just take and substitute the values ​​in the formula - the answer will be wrong. Even in terms of units of measurement, our answer “will not pass” the test for reasonableness. Compare:

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

And what happens if we do not translate into one system of measurement? The answer has a strange dimension and % is an incorrect result.

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

Length units:

centimeter = millimeters

decimeter = centimeters = millimeters

meter = decimeters = centimeters = millimeters

kilometer = meters

Time units:

minute = seconds

hour = minutes = seconds

days = hours = minutes = seconds

Advice: When converting units of measurement related to time (minutes to hours, hours to seconds, etc.), imagine a clock face in your head. It can be seen with the naked eye that minutes is 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 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 minute from an hour (mentally imagined a clock face, and said that minutes is a quarter of an hour), respectively - min \u003d h.

Intelligence.

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, reasonableness, that's about it)

Calculation.

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

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

Far from always, the tasks for movement are as simple as we solved before. Very often, in order to correctly solve a problem, you need to not just draw a competent drawing, but also make 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 miles per hour than a cyclist.

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

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

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

Kind of empty, right? We draw a table.

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

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 ​​\u200b\u200bin there. You remember how important this is, right?

Do you have a table like this?

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

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

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

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

The table has changed. We have left not filled only one column - time. How to find the time when there is a path and speed?

That's right, divide the path by the speed. Enter it in the table.

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

What can we reflect on it?

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

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

What data is not shown in the table or in the figure?

Right. The time by which the motorcyclist arrived earlier than the cyclist. We know that the time difference 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: writing and solving equations - manipulations that lead to the only correct answer.

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

Compilation of the equation:

Look at your table, at the last condition that was not included in it, and think about the relationship between what and what can we put 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 time of the motorcyclist from his time, we will just get the difference given to us.

This equation is rational. If you don't know what it is, read the topic "".

We bring the terms to a common denominator:

Let's open the brackets and give like terms: Phew! Got it? Try your hand at the next task.

Equation solution:

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 responses. Look what we got for? That's right, the speed of the cyclist.

We recall the rule "3P", more specifically "reasonableness". Do you understand what I mean? Exactly! Speed ​​cannot be negative, so our answer is km/h.

Second task

Two cyclists set out on a 1-kilometer run at the same time. The first one was driving at a speed that was 1 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 to the finish line second. Give your answer in km/h.

I recall the solution algorithm:

• Read the problem a couple of times - learn all the details. Got it?
• Start drawing the drawing - in which direction are they moving? how far did they travel? Did you draw?
• Check if all the quantities you have are of the same dimension and start writing out the condition of the problem briefly, making up a table (do you remember what columns are there?).
• While writing all this, think about what to take for? Chose? Record in the table! Well, now it’s simple: we make an equation and solve it. Yes, and finally - remember the "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

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

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

Got what I mean?

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

Read the questions carefully - 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.

Look at the dimension not only during the solution itself, but also when writing down the answers.

Tasks for movement in a circle

The bodies in the tasks may not necessarily move in a straight line, but also in a circle, for example, cyclists can ride along a circular track. Let's take a look at this problem.

Task #1

A cyclist left the point of the circular track. In minutes he had not yet returned to the checkpoint, and a motorcyclist followed him from the checkpoint. Minutes after departure, 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 track is km. Give your answer in km/h.

Solution of problem No. 1

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

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

But at the same time, the motorcyclist drove exactly one lap more, 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.

Got it? Try to solve the following problems yourself:

Tasks for independent work:

1. Two mo-to-tsik-li-hundreds start-to-tu-yut one-but-time-men-but in one-right-le-ni from two dia-met-ral-but pro-ty-in-po- false points of a circular route, the length of a swarm is equal to km. After how many minutes, mo-the-cycle-lists are equal for the first time, if the speed of one of them is by km / h more than the speed of the other th?
2. From one point of the circle-howl of the highway, the length of some swarm is equal to km, at the same time, in one right-le-ni, there are two motorcyclists. The speed of the first motorcycle is km / h, and minutes after the start, he was ahead of the second motorcycle by one lap. Find the speed of the second 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-li-hundred, then the speed of the second mo-to-cycle-li-hundred is km / h. Let the first time mo-the-cycle-lists be equal in hours. In order for mo-the-cycle-li-stas to be equal, the faster one must overcome them from the beginning distance, equal in lo-vi-not to the length of the route.

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

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

   The speed of the second motorcyclist is km/h.

Tasks for the course

Now that you're good at solving problems "on land", let's move on to the water and look at the scary problems associated with the current.

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

The current velocity in the lake is .

The raft will only move if you start rowing yourself. The speed he gains will be own speed of the raft. No matter where you swim - left, right, the raft will move at the same speed with which you row. It's clear? It's logical.

Now imagine that you are lowering the raft onto the river, turn away to take the rope ..., turn around, and he ... floated away ...

This happens because the river has a flow rate, 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 not rowing) - it moves with the speed of the current.

Got it?

Then answer this question - "How fast will the raft float on the river if you sit and row?" Thinking?

Two options are possible here.

Option 1 - you go with the flow.

And then you swim at your own speed + the speed of the current. The current 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 make more and more efforts 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 a mile. When will you cover this distance faster? When will you move with the flow or against?

Let's solve the problem and check.

Let's add to our path 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 current?

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

This is the whole essence of the tasks on flow with the flow.

Let's complicate the task a little.

Task #1

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

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

Solution of problem No. 1

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

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

What did we charge for?

Flow speed. Then this will be the answer :)

The speed of the current is km/h.

Task #2

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

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

Solution of problem No. 2

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

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

Converting this to hours:

hour minutes = h.

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

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

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

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

Did she swim all hours? Rereading the task.

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

h kayak really floated.

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

We open the brackets and give like terms. Next, we solve the resulting quadratic equation.

With this, I think you can also handle it on your own. What answer did you get? I have km/h.

Summing up

ADVANCED LEVEL

Movement tasks. Examples

Consider examples with solutionsfor each type of task.

moving with the flow

One of the simplest tasks tasks for the movement on the river. 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 in hours. Find the speed of the current if the speed of the boat in still water is km/h.

Solution #1:

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

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

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

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

So we can equate the distance. To do this, we immediately use distance formula:

Often it is necessary to use formula for time:

Example #2:

A boat travels a distance in km against the current for 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 write an equation. The time upstream is one hour longer than the time downstream.

It is written like this:

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

We got the usual rational equation, we 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 the bodies move towards each other;
• speed difference if the bodies are moving in the same direction.

Example #1

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

I solution way:

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

Solution 2:

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

In total, they traveled all km. Means,

Other motion tasks

Example #1:

A car left point A for point B. Simultaneously with it, another car left, which traveled exactly half the way at a speed of km/h less than the first one, and the second half of the way it drove 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 the time of the first car, and to the right - the second:

Simplify the expression on the right side:

We divide each term by AB:

It turned out the usual rational equation. Solving it, we get two roots:

Of these, only one is larger.

Answer: km/h.

Example #2

A cyclist left point A of the circular track. After a few minutes, he had not yet returned to point A, and a motorcyclist followed him from point A. Minutes after departure, 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 track 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 -.

In doing so, they traveled equal distances:

Between meetings, the cyclist traveled the distance, and the motorcyclist -. But at the same time, the motorcyclist drove exactly one lap more, 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 FORMULA

1. Basic formula

2. Relative motion

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

3. Move 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 have helped you deal with the tasks of movement...

Now it's your turn...

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

And this is already half way.

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

Which cause the greatest difficulty?

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

Write to us and good luck on your exams!

Solving problems on "movement on water" is difficult for many. There are several types of speeds in them, so the decisive ones start to get confused. To learn how to solve problems of this type, you need to know the definitions and formulas. The ability to draw up diagrams greatly facilitates the understanding of the problem, contributes to the correct compilation of the equation. A correctly composed equation is the most important thing in solving any type of problem.

Instruction

In the tasks "on the movement along the river" there are speeds: own speed (Vс), speed with the flow (Vflow), speed against the current (Vpr.flow), current speed (Vflow). It should be noted that the own speed of a watercraft is the speed in still water. To find the speed with the current, you need to add your own to the speed of the current. In order to find the speed against the current, it is necessary to subtract the speed of the current from the own speed.

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

Vac = Vc + Vac

Vpr. tech.=Vs-Vtech.

Vpr. flow = Vac. - 2Vtech.

Vac.=Vpr. tech+2Vtech

Vtech.=(Vstream. - Vpr.tech.)/2

Vc=(Vac.+Vc.flow)/2 or Vc=Vac.+Vc.

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

Example 1. Boat speed downstream is 21.8 km/h and upstream is 17.2 km/h. Find your own speed of the boat and the speed of the river.

Solution: According to the formulas: Vc \u003d (Vac. + Vpr.ch.) / 2 and Vch. \u003d (Vr. - Vpr.ch.) / 2, we find:

Vtech \u003d (21.8 - 17.2) / 2 \u003d 4.62 \u003d 2.3 (km / h)

Vc \u003d Vpr tech. + Vtech \u003d 17.2 + 2.3 \u003d 19.5 (km / h)

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

Example 2. The steamboat passed 24 km against the current and returned back, having spent 20 minutes less on the way back than when moving against the current. Find its own speed in still water if the current speed is 3 km/h.

For X we take the own speed of the ship. Let's make a table where we will enter all the data.

Against 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 trip than on the downstream trip, we compose and solve the equation.

20 min=1/3 hour.

24 / (X-3) - 24 / (X + 3) \u003d 1/3

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

72X+216-72X+216-X2+9=0

Х=21(km/h) – own speed of the steamer.

Answer: 21 km/h.

note

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

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

Purpose: to help students more fully master the technologies for solving problems on this topic.

Tasks for movement on water.

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

At first he did it himself, then rafts, boats, sailing ships appeared. With the development of technology, steamships, motor ships, 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.

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

In a puddle, the boat will stand still, and in a stream it will float, as the water in it "runs" to a lower place and carries it with it. The same will happen with a raft or a boat.

In the lake they will stand still, and in the river they will swim.

Consider the first option: a puddle and a lake. Water does not move in them and is called standing.

The boat will float in a 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. Such a movement is called movement in still water.

Is it different from driving on the road? Answer: no. And this means that we 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 motorboat sailed 60 km across the lake in 4 hours.

Find the own speed of the motorboat.

Answer: 15 km/h.

Task 3. How long will it take for a boat whose own speed is

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

Answer: 3 hours.

So, To find the distance traveled, you need to multiply the speed by the time.

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

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

Recall a paper boat in a stream. It floated because the water in it moves.

Such a movement is called downstream. 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 speed. (How to measure it?)

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

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

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

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

Those who do not know how to swim can imagine a situation where a strong wind is blowing. Consider two cases:

1) the wind blows in the back,

2) the wind blows in the face.

In both cases it is difficult to go. The wind in the back makes us run, which means that the speed of our movement increases. The wind in the face knocks us down, slows down. The speed is thus reduced.

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

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

Problem 5. The own speed of the boat 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 has to sail against the current of the river. Without a motor, or at least an oar, the current would carry her in the opposite direction. But, if you give the boat its own speed (start the engine or land a 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 own speed of the boat is 17 km/h.

Find the speed of the boat against the current.

Answer: 14 km/h.

Problem 7. The own speed of the ship is 47.2 km/h, and the speed of the river is 4.7 km/h. Find the speed of the boat upstream and downstream.

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 own speed of the boat 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 with the current if the speed of the river is 2.7 km/h.

Answer: 13.3 km/h; 16 km/h

Let us introduce the following notation:

V s. - own speed,

V tech. - flow speed,

V on current - flow speed,

V pr.tech. - speed against the current.

Then the following formulas can be written:

V no tech = V c + V tech;

V n.p. flow = V c - V flow;

Let's try to represent it graphically:

Conclusion: the difference in velocities downstream and upstream is equal to twice the current velocity.

Vno tech - Vnp. tech = 2 Vtech.

Vtech \u003d (V by tech - Vnp. tech): 2

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

Find the speed of the boat with the current.

Answer: 31 km/h.

2) The speed of a motorboat downstream 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 in 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.

According to the curriculum in mathematics, children are required to learn how to solve problems for movement in the original school. However, tasks of this type often cause difficulties for students. It is important that the child realizes what his own speed , speed flow, speed downstream and speed against the flow. Only under this condition will the student be able to easily solve problems for movement.

You will need

• Calculator, pen

Instruction

1. Own speed- This speed boats or other vehicles in static water. Designate it - V own. The water in the river is in motion. So she has her speed, which is called speed th current (V current) Designate the speed of the boat along the river as V along the current, and speed against the current - V pr. tech.

2. Now remember the formulas needed to solve problems for movement: V pr. tech. = V own. – V tech.V tech.= V own. + V tech.

3. It turns out, based on these formulas, it is possible to make the following results. If the boat moves against the flow of the river, then V own. = V pr. tech. + V tech. If the boat moves with the flow, then V own. = V according to current – V tech.

4. We will solve several problems for moving along the river. Task 1. The speed of the boat in spite of the flow of the river is 12.1 km / h. Discover your own speed boats, knowing that speed river flow 2 km / h. Solution: 12.1 + 2 \u003d 14, 1 (km / h) - own speed boats. Task 2. The speed of the boat along the river is 16.3 km / h, speed river current 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. Convert km/h to m/min: 14.4 / 0.06 = 240 (m/min.). This means that in 1 minute the boat would pass 240 m. Task 3. Two boats set off at the same time opposite each other from 2 points. The 1st boat moved along 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 current 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. 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 little trip down the river and the student will learn to “click puzzles like nuts”.

You will need

• Calculator, pen.

Instruction

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

2. In order to detect the speed of a ship moving 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 stagnant water, say, in a lake. Let's designate it - V own. The speed of the current is determined by how far the river carries the object per unit of time. Let's designate it - V tech.

3. In order to find the speed of the vessel moving against the current (V pr. tech.), It is necessary to subtract the speed of the current from the vessel's own speed. It turns out that we got the formula: V pr. tech. = V own. – V tech.

4. Let's find the speed of the ship against the flow of the river, if it is known that the own speed of the ship is 15.4 km / h, and the speed of the river is 3.2 km / h.15.4 - 3.2 \u003d 12.2 (km / h ) is the speed of the vessel moving against the current of the river.

5. In motion tasks, it is often necessary to convert km/h to m/s. In order to do this, it is necessary to remember that 1 km = 1000 m, 1 hour = 3600 s. Consequently, x km / h \u003d x * 1000 m / 3600 s \u003d x / 3.6 m / s. It turns out that in order to convert km / h to m / s, it is necessary to divide by 3.6. Let's say 72 km / h \u003d 72: 3.6 \u003d 20 m / s. In order to convert m / s to km / h, you must multiply by 3, 6. Let's say 30 m/s = 30 * 3.6 = 108 km/h.

6. Convert x km/h to m/min. To do this, recall that 1 km = 1000 m, 1 hour = 60 minutes. So x km/h = 1000 m / 60 min. = x / 0.06 m/min. Therefore, in order to convert km / h to m / min. must be divided by 0.06. Let's say 12 km/h = 200 m/min. In order 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
Do not forget about the units in which you measure the speed.

Note!
Do not forget about the units in which you measure the 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. In order to translate m / min. in km/h, multiply by 0.06.

Helpful advice
Drawing helps to solve the problem of movement.