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

You will need

• Calculator, pen

Instructions

Own speed- it speed boats or other means of transportation in still water. Designate it - V proper.
The water in the river is in motion. So she has her speed, which is called speed yu current (V flow.)
The speed of the boat along the river, denote - V along the river, and speed upstream - V pr. flow.

Now remember the formulas needed to solve motion problems:
V pr. Current = V proper. - V tech.
V on current = V own. + V current

So, based on these formulas, the following conclusions can be drawn.
If the boat is moving against the stream of the river, then V proper. = V pr. Flow. + V current
If the boat is moving with the current, then V proper. = V on flow. - V tech.

Let's solve several problems on the movement along the river.
Task 1. The speed of the boat against the river flow is 12.1 km / h. Find your own speed boats, knowing that speed river flow 2 km / h.
Solution: 12.1 + 2 = 14.1 (km / h) - own speed boats.
Task 2. The speed of the boat along the river is 16.3 km / h, speed river flow 1.9 km / h. How many meters would this boat go in 1 minute if it was in still water?
Solution: 16.3 - 1.9 = 14.4 (km / h) - own speed boats. Let's translate km / h into m / min: 14.4 / 0.06 = 240 (m / min.). This means that in 1 minute the boat would have covered 240 m.
Problem 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. speed each boat, if it is known that speed river flow 2 km / h.
Solution: 1) 42/3 = 14 (km / h) - speed movement along the river of the first boat.
2) 39/3 = 13 (km / h) - speed movement against the river flow of the second boat.
3) 14 - 2 = 12 (km / h) - own speed the first boat.
4) 13 + 2 = 15 (km / h) - own speed the second boat.

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.

Problems 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 it took to overcome it.

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

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

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

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

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

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

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

How far will the boat take in 3 hours?

Answer: 48 km.

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

Problem 2. The motor boat sailed 60 km across the lake in 4 hours.

Find your own speed boat.

Answer: 15 km / h.

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

is 28 km / h to swim 84 km on the lake?

Answer: 3 hours.

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

To find the speed, the path length must be divided by the time.

To find the time, the path length must be divided by the speed.

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

This movement is called downstream... And in the opposite direction - upstream.

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 carry

any object (sliver, 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 the river "helps" to swim in one direction, and "interferes" in the other.

Those who cannot swim can imagine a situation when a strong wind is blowing. Consider two cases:

1) the wind blows in the back,

2) the wind blows in the face.

And in either case 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 our face knocks us down, slows down. At the same time, the speed decreases.

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

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

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

Answer: 25 km / h.

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

So , in order to find the speed of the boat against the current, it is necessary to subtract the speed of the current from its own speed.

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

Find the speed of the boat upstream.

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 upstream.

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

Problem 8. The speed of the motor boat downstream is 12.4 km / h. Find your own boat speed if the river speed 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 your own boat speed and downstream speed if the river speed is 2.7 km / h.

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

Let us introduce the following notation:

V c. - own speed,

V tech. - current speed,

V on tech. - downstream speed,

V pr. Leak. - speed upstream.

Then you can write the following formulas:

V no flow = V c + V flow;

V np. flow = V c - V flow;

Let's try to depict this graphically:

Conclusion: the difference between the velocities upstream and upstream is equal to the doubled current velocity.

Vno tech - Vnp. flow = 2 Vflow.

Vflow = (Vflow - Vnp.flow): 2

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

Find the speed of the boat downstream.

Answer: 31 km / h.

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

Answer: 8 km / h.

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

* - when solving clause 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 mathematics curriculum, children are required to learn how to solve movement problems in their original school. However, tasks of this type often cause difficulties for students. It is important for the child to realize what his own speed , speed currents, speed downstream and speed contrary to the current. Only under this condition will the student be able to easily solve movement problems.

You will need

• Calculator, pen

Instructions

1. Own speed- it speed boats or other means of transportation in static water. Label it - V proper. The water in the river is in motion. So she has her speed which is called speed current (V current) Speed ​​of the boat along the river, denote - V along the current, and speed opposite the current - V pr. flow.

2. Now remember the formulas needed to solve traffic problems: V pr. Flow = V proper. - V current, V current = V own. + V current

3. It turns out, based on these formulas, it is allowed to make the following results: If the boat is moving against the flow of the river, then V proper. = V pr. Flow. + V current. If the boat is moving with the current, then V proper. = V on flow. - V tech.

4. Let's solve several problems on the movement along the river. Problem 1. The speed of the boat against the river flow is 12.1 km / h. Discover your own speed boats, knowing that speed river flow 2 km / h Solution: 12.1 + 2 = 14.1 (km / h) - own speed boats. Task 2. The speed of the boat along the river is 16.3 km / h, speed river flow 1.9 km / h. How many meters would this boat go in 1 minute if it was in still water? Solution: 16.3 - 1.9 = 14.4 (km / h) - own speed boats. Let's translate km / h into m / min: 14.4 / 0.06 = 240 (m / min.). This means that in 1 minute the boat would have covered 240 m. Problem 3. Two boats set off at the same time opposite each other from 2 points. The 1st boat was moving 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. speed any boat, if it is known that speed river flow 2 km / h Solution: 1) 42/3 = 14 (km / h) - speed movement along the river of the first boat. 2) 39/3 = 13 (km / h) - speed movement against the flow of the river of the second boat. 3) 14 - 2 = 12 (km / h) - own speed the first boat. 4) 13 + 2 = 15 (km / h) - own speed the second boat.

Movement problems seem difficult only at first glance. In order to discover, say, speed the movement of the vessel in spite of currents, it is enough to imagine the situation expressed in the problem. Take your child on a small trip along the river, and the student will learn to “click puzzles like nuts”.

You will need

• Calculator, pen.

Instructions

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

2. In order to detect the speed of movement of a vessel opposite the current, you need to know the own speed of the vessel and the speed of the current. The own speed is the speed of the vessel in still water, say, in a lake. Let's designate it - V proper. 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 movement of the vessel against the current (V pr. Flow), 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. Flow = V own. - V tech.

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

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

6. Let's translate x km / h into m / min. To do this, remember that 1 km = 1000 m, 1 h = 60 minutes. Hence, x km / h = 1000 m / 60 min. = x / 0.06 m / min. Consequently, in order to convert km / h to m / min. must be divided by 0.06. Let's say 12 km / h = 200 m / min. to translate m / min. in km / h must be multiplied by 0.06, say 250 m / min. = 15 km / h

Useful 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, divide by 3.6. To convert m / s to km / h, 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 must be multiplied by 0.06.

Useful advice
Drawing helps to solve the problem of movement.

Many people find it difficult to solve problems on "movement on water". 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 makes it very easy to understand the problem, contributes to the correct drawing up of the equation. And a well-formed equation is the most important thing in solving any type of problem.

Instructions

In the problems "on the movement along the river" there are speeds: own speed (Vс), speed along the current (Vcircuit), speed upstream (Vpr. 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 its own speed.

The first thing that you need to learn and know "by the teeth" - the formulas. Write down and remember:

Vin flow = Vc + Vflow.

Vpr. flow = Vc-V flow

Vpr. flow = V flow. - 2V leak.

Vreq. = Vpr. flow + 2V

Vflow = (Vflow - Vflow) / 2

Vc = (Vcircuit + Vcr.) / 2 or Vc = Vcr. + Vcr.

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

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

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

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

Vc = Vpr flow + Vflow = 17.2 + 2.3 = 19.5 (km / h)

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

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

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

Against the flow. With the flow

Distance 24 24

Speed ​​X-3 X + 3

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

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

20 minutes = 1/3 hours.

24 / (X-3) - 24 / (X + 3) = 1/3

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

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

X = 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 body of water.