Movement towards each other. Tasks for oncoming traffic (finding time and speed)

Mathematics is a rather difficult subject, but absolutely everyone will have to pass it in the school course. Movement tasks are especially difficult for students. How to solve without problems and a lot of time spent, we will consider in this article.

Note that if you practice, then these tasks will not cause any difficulties. The decision process can be developed to automatism.

Varieties

What is meant by this type of task? These are quite simple and simple tasks, which include the following varieties:

• oncoming traffic;
• after;
• movement in the opposite direction;
• river movement.

We propose to consider each option separately. Of course, we will analyze only on examples. But before we move on to the question as to movement, it is worth introducing one formula that we will need when solving absolutely all tasks of this type.

Formula: S=V*t. A little explanation: S is the path, the letter V denotes the speed of movement, and the letter t denotes time. All quantities can be expressed through this formula. Accordingly, speed is equal to distance divided by time, and time is distance divided by speed.

Movement towards

This is the most common type of task. To understand the essence of the solution, consider the following example. Condition: "Two friends on bicycles set off at the same time towards each other, while the path from one house to another is 100 km. What will be the distance after 120 minutes, if it is known that the speed of one is 20 km per hour, and the second is fifteen. " Let's move on to the question of how to solve the problem of the oncoming movement of cyclists.

To do this, we need to introduce another term: "speed of convergence". In our example, it will be equal to 35 km per hour (20 km per hour + 15 km per hour). This will be the first step in solving the problem. Next, we multiply the approach speed by two, since they moved for two hours: 35 * 2 = 70 km. We have found the distance that the cyclists will approach in 120 minutes. The last action remains: 100-70=30 kilometers. With this calculation, we found the distance between cyclists. Answer: 30 km.

If you do not understand how to solve the problem of oncoming traffic using the speed of approach, then use another option.

Second way

First, we find the path that the first cyclist traveled: 20*2=40 kilometers. Now the path of the 2nd friend: fifteen times two, which equals thirty kilometers. We add up the distance traveled by the first and second cyclists: 40+30=70 kilometers. We learned which path they covered together, so it remains to subtract the distance traveled from the entire path: 100-70 = 30 km. Answer: 30 km.

We have considered the first type of motion problem. How to solve them, now it’s clear, let’s move on to the next form.

Movement in the opposite direction

Condition: "Two hares galloped out of the same hole in the opposite direction. The speed of the first is 40 km per hour, and the second is 45 km per hour. How far will they be from each other in two hours?"

Here, as in the previous example, there are two possible solutions. In the first, we will proceed in the usual way:

1. Path of the first hare: 40*2=80 km.
2. Path of the second hare: 45*2=90 km.
3. The path they traveled together: 80+90=170 km. Answer: 170 km.

But another option is also possible.

Removal speed

As you may have guessed, in this task, similarly to the first, a new term will appear. Consider the following type of motion problem, how to solve them using the removal rate.

We will find it first of all: 40 + 45 = 85 kilometers per hour. It remains to find out what is the distance separating them, since all other data are already known: 85 * 2 = 170 km. Answer: 170 km. We considered the solution of problems on the movement traditional way, as well as using the speed of approach and removal.

Chasing after

Let's look at an example of a problem and try to solve it together. Condition: "Two schoolchildren, Kirill and Anton, left the school and moved at a speed of 50 meters per minute. Kostya followed them six minutes later at a speed of 80 meters per minute. How long will Kostya catch up with Kirill and Anton?"

So, how to solve problems for moving after? Here we need the speed of convergence. Only in this case it is worth not adding, but subtracting: 80-50 \u003d 30 m per minute. In the second step, we find out how many meters separate the schoolchildren before Kostya leaves. For this 50 * 6 = 300 meters. The last action is to find the time during which Kostya will catch up with Kirill and Anton. To do this, the path of 300 meters must be divided by the approach speed of 30 meters per minute: 300:30=10 minutes. Answer: in 10 minutes.

findings

Based on the foregoing, some conclusions can be drawn:

• when solving motion problems, it is convenient to use the speed of approach and removal;
• if we are talking about oncoming movement or movement from each other, then these quantities are found by adding the speeds of objects;
• if we are faced with the task of moving after, then we use the action, the opposite of addition, that is, subtraction.

We have considered some problems for movement, how to solve them, figured it out, got acquainted with the concepts of "speed of approach" and "speed of removal", it remains to consider the last point, namely: how to solve problems for movement along the river?

Flow

Here you can meet again:

• tasks to move towards each other;
• pursuit movement;
• movement in the opposite direction.

But unlike the previous tasks, the river has a flow rate that should not be ignored. Here the objects will move either along the river - then this speed should be added to the own speed of the objects, or against the current - it must be subtracted from the speed of the object.

An example of a task for moving along a river

Condition: went downstream at a speed of 120 km per hour and returned back, while spending two hours less time than against the current. What is the speed jet ski in stagnant water?" We are given a current velocity of one kilometer per hour.

Let's move on to the solution. We propose to create a table for good example. Let's take the speed of a motorcycle in still water as x, then the speed downstream is x + 1, and against x-1. The round trip distance is 120 km. It turns out that the time spent moving upstream is 120:(x-1), and downstream 120:(x+1). It is known that 120:(x-1) is two hours less than 120:(x+1). Now we can proceed to filling in the table.

What we have: (120/(x-1))-2=120/(x+1) Multiply each part by (x+1)(x-1);

120(x+1)-2(x+1)(x-1)-120(x-1)=0;

We solve the equation:

We notice that there are two answers here: + -11, since both -11 and +11 give 121 squared. But our answer will be positive, since the speed of the motorcycle cannot have a negative value, therefore, we can write the answer: 11 km per hour . Thus, we have found the required quantity, namely the speed in still water.

We've covered everything possible options tasks for movement, now when solving them you should not have problems and difficulties. To solve them, you need to learn the basic formula and concepts such as "the speed of approach and removal." Be patient, work through these tasks, and success will come.

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Lesson content

Distance/speed/time problem

Task 1. The car is moving at a speed of 80 km/h. How many kilometers will he travel in 3 hours?

Decision

If a car travels 80 kilometers in one hour, then in 3 hours it will travel three times as much. To find the distance, you need to multiply the speed of the car (80 km / h) by the time of movement (3 hours)

80 × 3 = 240 km

Answer: A car travels 240 kilometers in 3 hours.

Task 2. A car travels 180 km in 3 hours at the same speed. What is the speed of the car?

Decision

Velocity is the distance traveled by a body per unit of time. A unit means 1 hour, 1 minute or 1 second.

If in 3 hours the car traveled 180 kilometers at the same speed, then dividing 180 km by 3 hours we will determine the distance that the car traveled in one hour. And this is the speed of movement. To determine the speed, you need to divide the distance traveled by the time of movement:

180: 3 = 60 km/h

Answer: car speed is 60 km/h

Task 3. A car traveled 96 km in 2 hours and a cyclist traveled 72 km in 6 hours. How many times was the car faster than the cyclist?

Decision

Let's determine the speed of the car. To do this, we divide the distance traveled by him (96 km) by the time of his movement (2 hours)

96: 2 = 48 km/h

Determine the speed of the cyclist. To do this, we divide the distance traveled by him (72 km) by the time of his movement (6 hours)

72: 6 = 12 km/h

Find out how many times the car was moving faster than the cyclist. To do this, we find the ratio 48 to 12

Answer: the car was moving 4 times faster than the cyclist.

Task 4. The helicopter covered a distance of 600 km at a speed of 120 km/h. How long was he in flight?

Decision

If in 1 hour the helicopter covered 120 kilometers, then having learned how many such 120 kilometers are in 600 kilometers, we will determine how long it was in flight. To find the time, you need to divide the distance traveled by the speed of movement.

600: 120 = 5 hours

Answer: the helicopter was on the way for 5 hours.

Task 5. The helicopter flew for 6 hours at a speed of 160 km/h. How far did he travel during this time?

Decision

If in 1 hour the helicopter covered 160 km, then in 6 hours it covered six times more. To determine the distance, you need to multiply the speed of movement by time

160 × 6 = 960 km

Answer: in 6 hours the helicopter covered 960 km.

Task 6. The distance from Perm to Kazan, equal to 723 km, was covered by the car in 13 hours. For the first 9 hours he drove at a speed of 55 km/h. Determine the speed of the car in the remaining time.

Decision

Determine how many kilometers the car traveled in the first 9 hours. To do this, multiply the speed with which he drove the first nine hours (55 km / h) by 9

55 × 9 = 495 km

Let's figure out how far to go. To do this, subtract from the total distance (723 km) the distance traveled in the first 9 hours of movement

723 − 495 = 228 km

The car drove these 228 kilometers in the remaining 4 hours. To determine the speed of the car in the remaining time, you need to divide 228 kilometers by 4 hours:

228: 4 = 57 km/h

Answer: vehicle speed for the remaining time was 57 km/h

Approach speed

The speed of approach is the distance traveled by two objects towards each other per unit of time.

For example, if two pedestrians start towards each other from two points, and the speed of the first one is 100 m/m, and the second one is 105 m/m, then the approach speed will be 100+105, i.e. 205 m/m. This means that every minute the distance between pedestrians will decrease by 205 meters.

To find the speed of approach, you need to add the velocities of the objects.

Suppose the pedestrians meet three minutes after the start of the movement. Knowing that they met in three minutes, we can find out the distance between the two points.

Every minute pedestrians covered a distance equal to two hundred and five meters. After 3 minutes they met. So, multiplying the speed of approach by the time of movement, we can determine the distance between two points:

205 × 3 = 615 meters

You can also define the distance between points in another way. To do this, find the distance traveled by each pedestrian before the meeting.

So, the first pedestrian walked at a speed of 100 meters per minute. The meeting took place in three minutes, which means that in 3 minutes he walked 100 × 3 meters

100 × 3 = 300 meters

And the second pedestrian walked at a speed of 105 meters per minute. In three minutes he walked 105 × 3 meters

105 × 3 = 315 meters

Now you can add the results and thus determine the distance between the two points:

300 m + 315 m = 615 m

Task 1. Two cyclists left two settlements towards each other at the same time. The speed of the first cyclist is 10 km/h and the speed of the second is 12 km/h. After 2 hours they met. Determine the distance between settlements

Decision

Find the speed of convergence of cyclists

10 km/h + 12 km/h = 22 km/h

Determine the distance between settlements. To do this, multiply the speed of approach by the time of movement

22 × 2 = 44 km

Let's solve this problem in the second way. To do this, we find the distances traveled by cyclists and add up the results.

Find the distance traveled by the first cyclist:

10 × 2 = 20 km

Find the distance traveled by the second cyclist:

12 × 2 = 24 km

Let's sum up the obtained distances:

20 km + 24 km = 44 km

Answer: The distance between settlements is 44 km.

Task 2. From two settlements, the distance between which is 60 km, two cyclists left at the same time towards each other. The speed of the first cyclist is 14 km/h and the speed of the second is 16 km/h. How many hours later did they meet?

Decision

Find the speed of convergence of cyclists:

14 km/h + 16 km/h = 30 km/h

In one hour, the distance between cyclists decreases by 30 kilometers. To determine how many hours they will meet, you need to divide the distance between settlements by the speed of convergence:

60:30 = 2 hours

So the cyclists met in two hours

Answer: cyclists met after 2 hours.

Task 3. From two settlements, the distance between which is 56 km, two cyclists left at the same time towards each other. They met two hours later. The first cyclist was traveling at a speed of 12 km/h. Determine the speed of the second cyclist.

Decision

Determine the distance traveled by the first cyclist. Like the second cyclist, he spent 2 hours on the way. Multiplying the speed of the first cyclist by 2 hours, we can find out how many kilometers he traveled before the meeting

12 × 2 = 24 km

In two hours the first cyclist traveled 24 km. In one hour, he walked 24:2, that is, 12 km. Let's graph it

Subtract from the total distance (56 km) the distance traveled by the first cyclist (24 km). So we determine how many kilometers the second cyclist traveled:

56 km − 24 km = 32 km

The second cyclist, like the first, spent 2 hours on the road. If we divide the distance he traveled by 2 hours, then we find out how fast he moved:

32: 2 = 16 km/h

So the speed of the second cyclist is 16 km/h.

Answer: the speed of the second cyclist is 16 km/h.

Removal speed

Removal velocity is the distance that increases per unit of time between two objects moving in opposite directions.

For example, if two pedestrians start from the same point in opposite directions, with the speed of the first being 4 km/h and the speed of the second 6 km/h, then the removal speed will be 4+6, i.e. 10 km/h. Every hour the distance between two pedestrians will increase by 10 kilometers.

To find the removal speed, you need to add the speeds of the objects.

So, for the first hour, the distance between pedestrians will be 10 kilometers. The following figure shows how this happens.

It can be seen that the first pedestrian walked his 4 kilometers in the first hour. The second pedestrian also walked his 6 kilometers in the first hour. In total, in the first hour, the distance between them became 4 + 6, that is, 10 kilometers.

After two hours, the distance between pedestrians will be 10 × 2, that is, 20 kilometers. The following figure shows how this happens:

Task 1. From one station, a freight train and a passenger express set off simultaneously in opposite directions. The speed of a freight train was 40 km/h, the speed of an express train was 180 km/h. What is the distance between these trains after 2 hours?

Decision

Let us determine the speed of trains removal. To do this, add their speeds:

40 + 180 = 220 km/h

We got the train removal speed equal to 220 km/h. This speed shows that in an hour the distance between trains will increase by 220 kilometers. To find out what distance will be between trains in two hours, you need to multiply 220 by 2

220 × 2 = 440 km

Answer: after 2 hours the distance between the trains will be 440 kilometers.

Task 2. A cyclist and a motorcyclist left the point simultaneously in opposite directions. The speed of a cyclist is 16 km/h and the speed of a motorcyclist is 40 km/h. What is the distance between the cyclist and the motorcyclist after 2 hours?

Decision

16 km/h + 40 km/h = 56 km/h

Determine the distance that will be between the cyclist and the motorcyclist after 2 hours. To do this, we multiply the removal speed (56 km / h) by 2 hours

56 × 2 = 112 km

Answer: after 2 hours the distance between the cyclist and the motorcyclist will be 112 km.

Task 3. A cyclist and a motorcyclist left the point simultaneously in opposite directions. The speed of a cyclist is 10 km/h and the speed of a motorcyclist is 30 km/h. In how many hours will the distance between them be 80 km?

Decision

Let us determine the removal speed of the cyclist and motorcyclist. To do this, add their speeds:

10 km/h + 30 km/h = 40 km/h

In one hour, the distance between a cyclist and a motorcyclist increases by 40 kilometers. To find out after how many hours the distance between them will be 80 km, you need to determine how many times 80 km contains 40 km each

80: 40 = 2

Answer: 2 hours after the start of the movement, there will be 80 kilometers between the cyclist and the motorcyclist.

Task 4. A cyclist and a motorcyclist left the point simultaneously in opposite directions. After 2 hours the distance between them was 90 km. The speed of the cyclist was 15 km/h. Determine the speed of the motorcyclist

Decision

Determine the distance traveled by the cyclist in 2 hours. To do this, multiply its speed (15 km / h) by 2 hours

15 × 2 = 30 km

The figure shows that the cyclist traveled 15 kilometers in every hour. In total, he walked 30 kilometers in two hours.

Subtract from the total distance (90 km) the distance traveled by the cyclist (30 km). So we will determine how many kilometers the motorcyclist has traveled:

90 km − 30 km = 60 km

A motorcyclist traveled 60 kilometers in two hours. If we divide the distance he traveled by 2 hours, then we find out how fast he moved:

60: 2 = 30 km/h

So the speed of the motorcyclist was 30 km/h.

Answer: the speed of the motorcyclist was 30 km/h.

The task of moving objects in one direction

In the previous topic, we considered problems in which objects (people, cars, boats) moved either towards each other or in opposite directions. At the same time, we found various distances, which have changed between objects over time. These distances were either approach speeds or removal rates.

In the first case, we found approach speed- in a situation where two objects moved towards each other. For a unit of time, the distance between objects decreased by a certain distance

In the second case, we found the removal velocity - in a situation where two objects were moving in opposite directions. For a unit of time, the distance between objects increased by a certain distance

But objects can also move in the same direction, and with different speed. For example, a cyclist and a motorcyclist can leave the same point at the same time, and the speed of the cyclist can be 20 kilometers per hour, and the speed of the motorcyclist is 40 kilometers per hour.

The figure shows that the motorcyclist is twenty kilometers ahead of the cyclist. This is due to the fact that in an hour he overcomes 20 kilometers more than a cyclist. Therefore, every hour the distance between a cyclist and a motorcyclist will increase by twenty kilometers.

In this case, 20 km/h is the speed at which the motorcyclist is moving away from the cyclist.

After two hours, the distance traveled by the cyclist will be 40 km. The motorcyclist will travel 80 km, moving away from the cyclist for another twenty kilometers - the total distance between them will be 40 kilometers

To find the speed of removal when moving in one direction, you need to subtract the lower speed from the greater speed.

In the example above, the removal speed is 20 km/h. It can be found by subtracting the speed of the cyclist from the speed of the motorcyclist. The speed of the cyclist was 20 km/h and the speed of the motorcyclist was 40 km/h. The speed of the motorcyclist is greater, so subtract 20 from 40

40 km/h − 20 km/h = 20 km/h

Task 1. We left the city in the same direction a car and bus. The speed of a car is 120 km/h and the speed of a bus is 80 km/h. How far apart will they be after 1 hour? 2 hours?

Decision

Let's find the removal rate. To do this, subtract the smaller speed from the greater speed

120 km/h − 80 km/h = 40 km/h

Every hour, a passenger car moves away from the bus by 40 kilometers. In one hour, the distance between the car and the bus will be 40 km. For 2 hours twice as much:

40 × 2 = 80 km

Answer: after one hour, the distance between the car and the bus will be 40 km, after two hours - 80 km.

Consider a situation in which the objects started their movement from different points, but in the same direction.

Let there be a house, a school and an attraction. From home to school 700 meters

Two pedestrians went to the attraction at the same time. And the first pedestrian went to the attraction from the house at a speed of 100 meters per minute, and the second pedestrian went to the attraction from school at a speed of 80 meters per minute. What is the distance between pedestrians after 2 minutes? In how many minutes after the start of the movement will the first pedestrian catch up with the second?

Let's answer the first question of the problem - what is the distance between pedestrians after 2 minutes?

Determine the distance traveled by the first pedestrian in 2 minutes. He was moving at a speed of 100 meters per minute. In two minutes, he will travel twice as much, that is, 200 meters.

100 × 2 = 200 meters

Determine the distance traveled by the second pedestrian in 2 minutes. He was moving at a speed of 80 meters per minute. In two minutes, he will go twice as much, that is, 160 meters

80 × 2 = 160 meters

Now we need to find the distance between the pedestrians

To find the distance between pedestrians, you can add the distance traveled by the second pedestrian (160m) to the distance from home to school (700m) and subtract the distance traveled by the first pedestrian (200m) from the result obtained.

700 m + 160 m = 860 m

860 m − 200 m = 660 m

Or from the distance from home to school (700m) subtract the distance traveled by the first pedestrian (200m) and add the distance traveled by the second pedestrian (160m) to the result

700 m − 200 m = 500 m

500 m + 160 m = 660 m

Thus, after two minutes, the distance between pedestrians will be 660 meters.

Let's try to answer the following question of the problem: in how many minutes after the start of movement will the first pedestrian catch up with the second?

Let's see what the situation was at the very beginning of the journey - when pedestrians had not yet begun their movement

As can be seen in the figure, the distance between pedestrians at the beginning of the journey was 700 meters. But already a minute after the start of movement, the distance between them will be 680 meters, since the first pedestrian moves 20 meters faster than the second:

100 m × 1 = 100 m

80 m × 1 = 80 m

700 m + 80 m − 100 m = 780 m − 100 m = 680 m

Two minutes after the start of the movement, the distance will decrease by another 20 meters and will be 660 meters. This was our answer to the first question of the problem:

100 m × 2 = 200 m

80 m × 2 = 160 m

700 m + 160 m − 200 m = 860 m − 200 m = 660 m

After three minutes, the distance will decrease by another 20 meters and will already be 640 meters:

100 m × 3 = 300 m

80 m × 3 = 240 m

700 m + 240 m − 300 m = 940 m − 300 m = 640 m

We see that with every minute the first pedestrian will approach the second by 20 meters, and eventually will catch up with him. We can say that the speed equal to twenty meters per minute is the speed of convergence of pedestrians. The rules for finding the speed of approach and removal when moving in the same direction are identical.

To find the approach speed when moving in one direction, you need to subtract the smaller one from the greater speed.

And since the original 700 meters decrease by the same 20 meters every minute, then we can find out how many times 700 meters contain 20 meters, thereby determining how many minutes the first pedestrian will catch up with the second

700: 20 = 35

So 35 minutes after the start of the movement, the first pedestrian will catch up with the second. For interest, we find out how many meters each pedestrian has walked by this time. The first one was moving at a speed of 100 meters per minute. In 35 minutes he walked 35 times more

100 × 35 = 3500 m

The second walked at a speed of 80 meters per minute. In 35 minutes he walked 35 times more

80 × 35 = 2800 m

The first one went 3500 meters and the second one 2800 meters. The first one went 700 meters more as it walked from the house. If we subtract these 700 meters from 3500, then we get 2800 m

Let's consider a situation in which objects move in one direction, but one of the objects started its movement before the other.

Let there be a house and a school. The first pedestrian went to school at a speed of 80 meters per minute. After 5 minutes, a second pedestrian followed him to the school at a speed of 100 meters per minute. In how many minutes will the second pedestrian overtake the first?

The second pedestrian started his movement in 5 minutes. By this time, the first pedestrian had already moved away from him at some distance. Let's find this distance. To do this, multiply its speed (80 m/m) by 5 minutes

80 × 5 = 400 meters

The first pedestrian moved away from the second by 400 meters. Therefore, at the moment when the second pedestrian starts his movement, there will be these same 400 meters between them.

But the second pedestrian is moving at a speed of 100 meters per minute. That is, it moves 20 meters faster than the first pedestrian, which means that with every minute the distance between them will decrease by 20 meters. Our task is to find out in how many minutes this will happen.

For example, in a minute the distance between pedestrians will be 380 meters. The first pedestrian will walk another 80 meters to his 400 meters, and the second will walk 100 meters

The principle here is the same as in the previous problem. The distance between pedestrians at the time of movement of the second pedestrian must be divided by the speed of convergence of pedestrians. The approach speed in this case is twenty meters. Therefore, to determine in how many minutes the second pedestrian will catch up with the first, you need to divide 400 meters by 20

400: 20 = 20

So in 20 minutes the second pedestrian will catch up with the first.

Task 2. From two villages, the distance between which is 40 km, a bus and a cyclist left at the same time in the same direction. The speed of a cyclist is 15 km/h and the speed of a bus is 35 km/h. In how many hours will the bus overtake the cyclist?

Decision

Let's find the speed of approach

35 km/h − 15 km/h = 20 km/h

Determine in hours the bus will catch up with the cyclist

40: 20 = 2

Answer: the bus will catch up with the cyclist in 2 hours.

The task of moving along the river

Vessels move along the river at different speeds. At the same time, they can move both with the flow of the river and against the flow. Depending on how they move (upstream or downstream), the speed will change.

Suppose the speed of the river is 3 km/h. If you lower a boat into a river, the river will carry the boat away at a speed of 3 km/h.

If you lower the boat into stagnant water, in which there is no current, then the boat will also stand. The speed of the boat in this case will be equal to zero.

If a boat floats on stagnant water in which there is no current, then they say that the boat is sailing with own speed.

For example, if powerboat floats on still water at a speed of 40 km/h, then they say that own speed of the boat is 40 km/h.

How to determine the speed of a ship?

If the ship is following the current of the river, then the speed of the river must be added to the own speed of the ship.

with the flow rivers, and the speed of the river is 2 km/h, then the speed of the river (2 km/h) must be added to the own speed of the motor boat (30 km/h)

30 km/h + 2 km/h = 32 km/h

The flow of the river can be said to help the motor boat with an additional speed equal to two kilometers per hour.

If the ship is sailing against the current of the river, then the speed of the current of the river must be subtracted from the own speed of the ship.

For example, if a motorboat is traveling at a speed of 30 km/h against the stream rivers, and the speed of the river is 2 km/h, then the speed of the river (2 km/h) must be subtracted from the own speed of the motorboat (30 km/h)

30 km/h − 2 km/h = 28 km/h

The current of the river in this case prevents the motor boat from moving freely forward, reducing its speed by two kilometers per hour.

Task 1. The speed of the boat is 40 km/h and the speed of the river is 3 km/h. How fast will the boat move down the river? Against the current of the river?

Answer:

If the boat moves along the current of the river, then its speed will be 40 + 3, that is, 43 km / h.

If the boat moves against the current of the river, then its speed will be 40 - 3, that is, 37 km / h.

Task 2. The speed of the ship in still water is 23 km/h. The speed of the river is 3 km/h. How far will the ship travel in 3 hours along the river? Against the stream?

Decision

The own speed of the ship is 23 km/h. If the ship moves along the river, then its speed will be 23 + 3, that is, 26 km / h. In three hours, he will travel three times as much

26 × 3 = 78 km

If the ship moves against the current of the river, then its speed will be 23 - 3, that is, 20 km / h. In three hours, he will travel three times as much

20 × 3 = 60 km

Task 3. The boat covered the distance from point A to point B in 3 hours 20 minutes, and the distance from point B to A in 2 hours 50 minutes. In which direction does the river flow: from A to B or from B to A, if it is known that the speed of the yacht did not change?

Decision

The speed of the yacht did not change. We will find out on which path she spent more time: on the path from A to B or on the path from B to A. The path that spent more time will be the path whose river flow went against the yacht

3 hours 20 minutes is longer than 2 hours 50 minutes. This means that the current of the river reduced the speed of the yacht and this was reflected in the travel time. 3 hours 20 minutes is the time taken to travel from A to B. So the river flows from point B to point A

Task 4. How long does it take to move against the current of a river?
the ship will travel 204 km if its own speed is
15 km / h, and the current speed is 5 times less than its own
ship speed?

Decision

It is required to find the time during which the ship will travel 204 kilometers against the river current. The own speed of the ship is 15 km/h. It moves against the current of the river, so you need to determine its speed with such a movement.

To determine the speed against the current of the river, you need to subtract the speed of the river from the own speed of the ship (15 km / h). The condition says that the speed of the river is 5 times less than the own speed of the ship, so first we determine the speed of the river. To do this, we reduce 15 km / h five times

15: 5 = 3 km/h

The speed of the river is 3 km/h. Subtract this speed from the speed of the ship

15 km/h − 3 km/h = 12 km/h

Now we determine the time for which the ship will cover 204 km at a speed of 12 km / h. The ship travels 12 kilometers per hour. To find out how many hours it will take him to cover 204 kilometers, you need to determine how many times 204 kilometers contains 12 kilometers each.

204: 12 = 17 h

Answer: the ship will cover 204 kilometers in 17 hours

Task 5. Moving along the river, in 6 hours the boat
walked 102 km. Determine your own speed of the boat,

Decision

Find out how fast the boat was moving along the river. To do this, the distance traveled (102 km) is divided by the time of movement (6 hours)

102: 6 = 17 km/h

Let us determine the own speed of the boat. To do this, from the speed at which she moved along the river (17 km / h), we subtract the speed of the river (4 km / h)

17 − 4 = 13 km/h

Task 6. Moving against the current of the river, in 5 hours the boat
walked 110 km. Determine your own speed of the boat,
if the current speed is 4 km/h.

Decision

Find out how fast the boat was moving along the river. To do this, the distance traveled (110 km) is divided by the time of movement (5 hours)

110: 5 = 22 km/h

Let us determine the own speed of the boat. The condition says that she was moving against the current of the river. The speed of the river flow was 4 km/h. This means that the own speed of the boat has been reduced by 4. Our task is to add these 4 km/h and find out the own speed of the boat

22 + 4 = 26 km/h

Answer: the boat's own speed is 26 km/h

Task 7. How long does it take for a boat to move upstream
travel 56 km if the current speed is 2 km/h and its
own speed 8 km/h more than the speed of the current?

Decision

Find the own speed of the boat. The condition says that it is 8 km / h more than the current speed. Therefore, to determine the own speed of the boat, we add another 8 km/h to the current speed (2 km/h).

2 km/h + 8 km/h = 10 km/h

The boat is moving against the current of the river, so from the own speed of the boat (10 km / h) we subtract the speed of the river (2 km / h)

10 km/h − 2 km/h = 8 km/h

Find out how long the boat will travel 56 km. To do this, we divide the distance (56 km) by the speed of the boat:

56:8 = 7h

Answer: when moving against the current of the river, the boat will cover 56 km in 7 hours

Tasks for independent solution

Task 1. How long will it take for a pedestrian to walk 20 km if his speed is 5 km/h?

Decision

In one hour, a pedestrian walks 5 kilometers. To determine how long it will take him to cover 20 km, you need to find out how many times 20 kilometers contain 5 km each. Or use the rule of finding time: divide the distance traveled by the speed of movement

20:5 = 4 hours

Task 2. From point BUT to paragraph AT A cyclist rode for 5 hours at a speed of 16 km/h, and he rode back along the same path at a speed of 10 km/h. How long did the cyclist take to get back?

Decision

Determine the distance from the point BUT to point AT. To do this, we multiply the speed at which the cyclist traveled from the point BUT to paragraph AT(16km/h) for driving time (5h)

16 × 5 = 80 km

Let's determine how much time the cyclist spent on the way back. To do this, the distance (80 km) is divided by the speed (10 km / h)

Problem 3. A cyclist rode for 6 hours at a certain speed. After he traveled another 11 km at the same speed, his path became equal to 83 km. How fast was the cyclist traveling?

Decision

Determine the distance traveled by the cyclist in 6 hours. To do this, from 83 km we subtract the path that he traveled after six hours of movement (11 km)

83 − 11 = 72 km

Determine how fast the cyclist rode for the first 6 hours. To do this, we divide 72 km by 6 hours

72: 6 = 12 km/h

Since the condition of the problem says that the cyclist traveled the remaining 11 km at the same speed as in the first 6 hours of movement, then the speed equal to 12 km / h is the answer to the problem.

Answer: A cyclist is traveling at a speed of 12 km/h.

Task 4. Moving against the current of the river, the ship covers a distance of 72 km in 4 hours, and the raft sails the same distance in 36 hours. How many hours will the ship cover a distance of 110 km if it floats along the river?

Decision

Find the speed of the river. The condition says that the raft can sail 72 kilometers in 36 hours. The raft cannot move against the current of the river. This means that the speed of the raft with which it overcomes these 72 kilometers is the speed of the river. To find this speed, you need to divide 72 kilometers by 36 hours.

72: 36 = 2 km/h

Find the own speed of the ship. First, we find the speed of its movement against the current of the river. To do this, we divide 72 kilometers by 4 hours

72: 4 = 18 km/h

If the speed of the ship against the river current is 18 km/h, then its own speed is 18 + 2, that is, 20 km/h. And along the river, its speed will be 20 + 2, that is, 22 km / h

By dividing 110 kilometers by the speed of the ship moving along the river (22 km / h), you can find out how many hours the ship will sail these 110 kilometers

Answer: the ship will travel 110 kilometers along the river for 5 hours.

Problem 5. Two cyclists left the same point at the same time in opposite directions. One of them was driving at a speed of 11 km/h, and the other one at a speed of 13 km/h. How far apart will they be after 4 hours?

21 × 6 = 126 km

Determine the distance traveled by the second ship. To do this, we multiply its speed (24 km / h) by the time it takes to meet (6 hours)

24 × 6 = 144 km

Determine the distance between the piers. To do this, add the distances traveled by the first and second ships

126 km + 144 km = 270 km

Answer: The first ship covered 126 km, the second - 144 km. The distance between the marinas is 270 km.

Problem 7. Two trains left Moscow and Ufa at the same time. After 16 hours they met. The Moscow train was moving at a speed of 51 km/h. How fast was the train leaving Ufa if the distance between Moscow and Ufa is 1520 km? What was the distance between the trains 5 hours after they met?

Decision

Let's determine how many kilometers the train that left Moscow passed before the meeting. To do this, multiply its speed (51 km / h) by 16 hours

51 × 16 = 816 km

We will find out how many kilometers the train left Ufa passed before the meeting. To do this, from the distance between Moscow and Ufa (1520 km) we subtract the distance traveled by the train that left Moscow

1520 − 816 = 704 km

Let's determine the speed with which the train was leaving Ufa. To do this, the distance traveled by him before the meeting must be divided by 16 hours

704: 16 = 44 km/h

Let's determine the distance that will be between the trains 5 hours after they meet. To do this, we find the speed of removal of trains and multiply this speed by 5

51 km/h + 44 km/h = 95 km/h

95 × 5 = 475 km.

Answer: A train leaving Ufa was moving at a speed of 44 km/h. In 5 hours after their meeting of trains, the distance between them will be 475 km.

Problem 8. Two buses set off from one point at the same time in opposite directions. The speed of one bus is 48 km/h, the other is 6 km/h faster. In how many hours will the distance between the buses be 510 km?

Decision

Find the speed of the second bus. It is 6 km/h more than the speed of the first bus

48 km/h + 6 km/h = 54 km/h

Let's find the speed of removal of buses. To do this, add their speeds:

48 km/h + 54 km/h = 102 km/h

In an hour, the distance between buses increases by 102 kilometers. To find out after how many hours the distance between them will be 510 km, you need to find out how many times 510 km contains 102 km / h

Answer: 510 km between buses will be in 5 hours.

Problem 9. The distance from Rostov-on-Don to Moscow is 1230 km. Two trains left Moscow and Rostov towards each other. The train from Moscow travels at a speed of 63 km/h, and the speed of the Rostov train is the speed of the Moscow train. At what distance from Rostov will the trains meet?

Decision

Find the speed of the Rostov train. It is the speed of the Moscow train. Therefore, to determine the speed of the Rostov train, you need to find from 63 km

63: 21 × 20 = 3 × 20 = 60 km/h

Find the speed of convergence of trains

63 km/h + 60 km/h = 123 km/h

Determine how many hours the trains will meet

1230: 123 = 10 h

We will find out at what distance from Rostov the trains will meet. To do this, it is enough to find the distance traveled by the Rostov train before the meeting

60 × 10 = 600 km.

Answer: the trains will meet at a distance of 600 km from Rostov.

Problem 10. From two piers, the distance between which is 75 km, two motor boats simultaneously departed towards each other. One was moving at a speed of 16 km / h, and the speed of the other was 75% of the speed of the first boat. How far apart will the boats be after 2 hours?

Decision

Find the speed of the second boat. It is 75% of the speed of the first boat. Therefore, to find the speed of the second boat, you need 75% of 16 km

16 × 0.75 = 12 km/h

Find the speed of approach of the boats

16 km/h + 12 km/h = 28 km/h

Every hour the distance between the boats will decrease by 28 km. After 2 hours, it will decrease by 28 × 2, that is, by 56 km. To find out what will be the distance between the boats at this moment, you need to subtract 56 km from 75 km

75 km − 56 km = 19 km

Answer: in 2 hours there will be 19 km between the boats.

Problem 11. A car with a speed of 62 km/h overtakes a truck with a speed of 47 km/h. After how much time and at what distance from the beginning of the movement will the passenger car catch up with the freight car, if the initial distance between them was 60 km?

Decision

Let's find the speed of approach

62 km/h − 47 km/h = 15 km/h

If initially the distance between the cars was 60 kilometers, then every hour this distance will decrease by 15 km, and in the end the passenger car will overtake the truck. To find out after how many hours this will happen, you need to determine how many times 60 km contains 15 km

Find out at what distance from the start of the movement the passenger car caught up with the truck. To do this, we multiply the speed of the passenger car (62 km / h) by the time of its movement until the meeting (4 hours)

62 × 4 = 248 km

Answer: the passenger car will catch up with the truck in 4 hours. At the time of the meeting, the passenger car will be at a distance of 248 km from the start of the movement.

Problem 12. Two motorcyclists left the same point in the same direction at the same time. The speed of one was 35 km/h, and the speed of the other was 80% of the speed of the first motorcyclist. How far apart will they be after 5 hours?

Decision

Find the speed of the second motorcyclist. It is 80% of the speed of the first motorcyclist. Therefore, to find the speed of the second motorcyclist, you need to find 80% of 35 km/h

35 × 0.80 = 28 km/h

The first rider moves 35-28 km/h faster

35 km/h − 28 km/h = 7 km/h

In one hour, the first motorcyclist overcomes 7 kilometers more. With every hour, she will approach the second motorcyclist for these 7 kilometers.

After 5 hours, the first motorcyclist will cover 35×5, i.e. 175 km, and the second motorcyclist will travel 28×5, i.e. 140 km. Let's determine the distance between them. To do this, subtract 140 km from 175 km

175 − 140 = 35 km

Answer: after 5 hours the distance between motorcyclists will be 35 km.

Problem 13. A motorcyclist whose speed is 43 km/h overtakes a cyclist whose speed is 13 km/h. In how many hours will the motorcyclist overtake the cyclist if the initial distance between them was 120 km?

Decision

Let's find the speed of approach:

43 km/h − 13 km/h = 30 km/h

If initially the distance between the motorcyclist and the cyclist was 120 kilometers, then every hour this distance will decrease by 30 km, and in the end the motorcyclist will catch up with the cyclist. To find out after how many hours this will happen, you need to determine how many times 120 km contains 30 km

So after 4 hours the motorcyclist will catch up with the cyclist

The figure shows the movement of a motorcyclist and a cyclist. It can be seen that 4 hours after the start of the movement, they leveled off.

Answer: The motorcyclist will overtake the cyclist in 4 hours.

Problem 14. A cyclist whose speed is 12 km/h overtakes a cyclist whose speed is 75% of his speed. After 6 hours, the second cyclist caught up with the cyclist riding first. What was the distance between cyclists initially?

Decision

Determine the speed of the cyclist in front. To do this, we find 75% of the speed of the cyclist riding behind:

12 × 0.75 \u003d 9 km / h - the speed of the person in front

Find out how many kilometers each cyclist traveled before the second caught up with the first:

12 × 6 \u003d 72 km - the driver behind drove
9 × 6 \u003d 54 km - the one in front drove

Find out what the distance was between the cyclists initially. To do this, from the distance traveled by the second cyclist (who was catching up), we subtract the distance traveled by the first cyclist (who was caught up)

It can be seen that the car is 12 km ahead of the bus.

To find out in how many hours the car will be ahead of the bus by 48 kilometers, you need to determine how many times 48 km contains 12 km each

Answer: 4 hours after the departure, the car will be 48 kilometers ahead of the bus.

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Motion problems with solutions

1. The first tourist traveled 2 hours on a bicycle at a speed of 16 km/h. After resting for 2 hours, he poisoned himself further at the same speed. 4 hours after the start of the cyclist, a second tourist on a motorcycle rode after him at a speed of 56 km/h. At what distance from the starting point will the motorcyclist overtake the cyclist?
2. Three cars set off from point A to point B one after another with an interval of 1 hour. The speed of the first car is 50 km/h, and the second - 60 km/h. Find the speed of the third car if it is known that it caught up with the first two cars at the same time.
3. The train was delayed on the way for 12 minutes, and then at a distance of 60 km made up for the lost time, increasing the speed by 15 km/h. Find the original speed of the train.
4. The distance between stations A and B is 103 km. A train left A for B and, having traveled some distance, was delayed, and therefore the path remaining to B passed at a speed 4 km/h greater than before. Find the initial speed of the train if it is known that the distance to B was 23 km longer than the distance traveled before the delay, and that the journey after the delay took 15 minutes more than the journey before it.
5. vehicle speed by flat area 5 km/h less than downhill speed and 15 km/h more than uphill speed. The road from A to B goes uphill and is 100 km long. Determine the speed of the car on a flat area, if the distance from A to B and back it traveled in 1 hour and 50 minutes.
6. The bus covers the distance between points A and B according to the schedule in 5 hours. Once, leaving A, the bus was delayed for 10 minutes 56 km from A and, in order to arrive at B on schedule, it had to most way to move at a speed exceeding the original by 2 km / h. Find the speed of the bus according to the schedule and the distance between points A and B, if it is known that this distance exceeds 100 km.
7. The train passes the platform in 32 seconds. In how many seconds will the train pass by a stationary observer if the length of the train is equal to the length of the platform?
8. Two trains set off towards each other at constant speeds, one from A to B, the other from B to A. They can meet in the middle of the journey if the train from A leaves 1.5 hours earlier. If both trains left at the same time, then after 6 hours the distance between them would be a tenth of the original. How many hours does each train spend traveling between A and B?
9. From pier A, a boat and a raft set off simultaneously downstream. The boat went down 96 km, then turned back and returned to A after 14 hours. Find the speed of the boat in still water and the speed of the current, if it is known that the boat met a raft on the way back at a distance of 24 km from A.
10. Point B is downstream of point A. At the same time, a raft and the first motorboat set sail from point A, and a second motorboat departed from point B. After some time, the boats met at point C, and during this time the raft sailed a third of the way from A to C. If the first boat sailed to point B without stopping, then the raft would arrive at point C in this time. If from point A the second boat sailed to point B, and the first boat sailed from point B to point A, then they would meet 40 km from point A. What is the speed of both boats in still water, as well as the distance between points A and B, if the speed of the river equal to 3 km/h?
11. Two bodies moving along a circle in the same direction meet every 112 minutes, and moving in opposite directions - every 16 minutes. In the second case, the distance between the bodies decreased from 40 m to 26 m in 12 s. How many meters per minute does each body travel and what is the circumference?
12. Two points, moving along a circle in the same direction, meet every 12 minutes, with the first going around the circle 10 seconds faster than the second. What part of the circle does each point cover in 1 s?
13. Two bodies move towards each other from two points, the distance between which is 390 m. The first body passed 6 m in the first second, and in each subsequent second it passed 6 m more than in the previous one. The second body moved uniformly with a speed of 12 m/s and started moving 5 s after the first one. How many seconds after the first body started moving, will they meet?

Tasks for independent solution

1. The road from A to D, 23 km long, goes first uphill, then on a flat area, and then downhill. A pedestrian, moving from A to D, covered the whole way in 5 hours and 48 minutes, and back, from D to A, in 6 hours and 12 minutes. The speed of its movement uphill is 3 km / h, on a flat area - 4 km / h, and downhill - 5 km / h. Determine the length of the road on a flat area. Answer: 8 km
2. At 5 o'clock in the morning, a mail train left station A in the direction of station B, 1080 km away from A. At 8 o'clock in the morning a passenger train left station B in the direction of A, which traveled 15 km more per hour than the postal train. When did the trains meet if their meeting took place in the middle of track AB? Answer: at 5 p.m.
3. Three cyclists set off from point A to point B. The first one was traveling at a speed of 12 km/h. The second set off 0.5 hours later than the first and traveled at a speed of 10 km/h. What is the speed of the third cyclist who set off 0.5 hours later than the second, if it is known that he caught up with the first 3 hours after he caught up with the second? Answer: 15 km/h
4. Two trains - a freight train 490 m long and a passenger train 210 m long - were moving towards each other along two parallel tracks. Driver passenger train noticed a freight train when he was 700 m away from it; 28 seconds later, the trains met. Determine the speed of each train if it is known that the freight train passes the traffic light 35 seconds slower than the passenger train. Answer: 36 km/h; 54 km/h
5. Tourist A and tourist B had to go out at the same time towards each other from the village M and the village N, respectively. However, tourist A was delayed and left 6 hours later. At the meeting, it turned out that A traveled 12 km less than B. Having rested, the tourists left the meeting place at the same time and continued their journey at the same speed. As a result, A came to village N 8 hours later, and B arrived to village M 9 hours after the meeting. Determine the distance MN and the speed of the tourists. Answer: 84 km; 6 km/h; 4 km/h
6. A pedestrian, a cyclist and a motorcyclist are moving along the highway in one direction at constant speeds. At that moment, when the pedestrian and the cyclist were at the same point, the motorcyclist was 6 km behind them, and at the moment when the motorcyclist caught up with the cyclist, the pedestrian was 3 km behind them. How many kilometers did the cyclist overtake the pedestrian at the moment when the pedestrian was overtaken by the motorcyclist? Answer: 2 km
7. Two tourists started simultaneously from point A to point B. The first tourist walked each kilometer 5 minutes faster than the second. Having traveled 20% of the distance from A to B, the first tourist turned back, came to A, stayed there for 10 minutes, went to B again and ended up there at the same time.
   but with a second tourist. Determine the distance from A to B if the second tourist traveled it in 2.5 hours. Answer: 10 km
8. The fisherman sailed on a boat from the pier against the current for 5 km and returned back to the pier. The speed of the river is 2.4 km/h. If the fisherman rowed with the same force in the still water of the lake on a boat with a sail that increases speed by 3 km / h, then he would swim 14 km in the same time. Find the speed of the boat in still water. Answer: 9.6 km/h
9. The motorboat sailed across the lake, and then went down the river flowing out of the lake. The distance traveled by the boat on the lake is 15% less than the distance traveled on the river. The boat travel time on the lake is 2% longer than on the river. By what percent is the speed of the boat down the river greater than the speed of the lake? Answer: 20%
10. A tourist sailed in a boat along the river from city A to city B and back, spending 10 hours on this. The distance between cities is 20 km. Find the speed of the current of the river, knowing that the tourist swam 2 km against the current of the river in the same time as 3 km with the current. Answer: 5/6 km/h
11. Two points move uniformly in the same direction along a circle 60 m long. One of them makes a complete revolution 5 s faster than the other. In this case, the coincidence of points occurs every time after 1 min. Determine the velocities of the points. Answer: 4 m/s; 3 m/s.
12. From points A and B, two bodies simultaneously began to move towards each other. The first one covered 1 m in the first minute, and 0.5 m more in each subsequent minute than in the previous one. The second body passed 6 m every minute. After how many minutes did both bodies meet if the distance between A and B is 117 m? Answer: after 12 minutes.
13. Two friends in the same boat swam along the river along the coast and returned along the same river route 5 hours after departure. The length of the entire flight was 10 km. According to their calculations, it turned out that for every 2 km upstream, on average, it took the same amount of time as for every 3 km downstream. Find the speed of the current of the river, as well as the travel time there and the travel time back. Answer: 5/12 km/h; 2 o'clock and 3 o'clock