vertical movement of the body. The motion of a body thrown vertically upwards

As we already know, gravity acts on all bodies that are on the surface of the Earth and near it. It doesn't matter if they are at rest or moving.

If a certain body falls freely to the Earth, then at the same time it will make uniformly accelerated motion, and the speed will increase constantly, since the velocity vector and the acceleration vector free fall will be aligned with each other.

The essence of the movement vertically upwards

If we toss a body vertically upwards, and at the same time, we assume that there is no air resistance, then we can assume that it also makes uniformly accelerated motion, with free fall acceleration, which is caused by gravity. Only in this case, the speed that we gave to the body during the throw will be directed upwards, and the acceleration of free fall is directed downwards, that is, they will be directed oppositely to each other. Therefore, the speed will gradually decrease.

After some time, the moment will come when the speed will be equal to zero. At this point, the body will reach its maximum height and stop for a moment. It is obvious that the greater the initial speed we give to the body, the greater the height it will rise by the time it stops.

Further, the body will begin to fall down with uniform acceleration, under the influence of gravity.

How to solve problems

When you come across tasks for the movement of the body upwards, which does not take into account air resistance and other forces, but it is believed that only gravity acts on the body, then since the movement is uniformly accelerated, you can apply the same formulas as for a rectilinear uniformly accelerated moving with some initial speed V0.

Since in this case the acceleration ax is the free fall acceleration of the body, ax is replaced by gx.

Vx=V0x+gx*t,
Sx=V(0x)*t+(gx*t^2)/2.

It should also be taken into account that when moving up, the gravitational acceleration vector is directed downwards, and the velocity vector is upwards, that is, they are oppositely directed, and therefore their projections will have different signs.

For example, if the Ox axis is directed upwards, then the projection of the velocity vector when moving upwards will be positive, and the projection of the gravitational acceleration will be negative. This must be taken into account when substituting values into formulas, otherwise a completely wrong result will be obtained.

The motion of a body thrown vertically upwards

I level. Read the text

If a certain body falls freely to the Earth, then it will perform uniformly accelerated motion, and the speed will increase constantly, since the velocity vector and the free fall acceleration vector will be co-directed with each other.

If we toss some body vertically upwards, and at the same time assume that there is no air resistance, then we can assume that it also makes uniformly accelerated motion, with free fall acceleration, which is caused by gravity. Only in this case, the speed that we gave to the body during the throw will be directed upwards, and the acceleration of free fall is directed downwards, that is, they will be directed oppositely to each other. Therefore, the speed will gradually decrease.

All formulas for uniformly accelerated motion applicable to the motion of a body thrown upwards. V0 always > 0

The motion of a body thrown vertically upwards is rectilinear movement with constant acceleration. If you direct the OY coordinate axis vertically upwards, aligning the origin of coordinates with the Earth's surface, then to analyze free fall without an initial velocity, you can use the formula https://pandia.ru/text/78/086/images/image002_13.gif" width="151 "height="57 src=">

Near the surface of the Earth, in the absence of a noticeable influence of the atmosphere, the speed of a body thrown vertically upwards changes in time according to a linear law: https://pandia.ru/text/78/086/images/image004_7.gif" width="55" height ="28">.

The speed of a body at a certain height h can be found by the formula:

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The height of the body for some time, knowing the final speed

https://pandia.ru/text/78/086/images/image008_5.gif" width="676" height="302 src=">

IIIlevel. Solve problems. For 9 b. 9a solves from the problem book!

1. A ball is thrown vertically upwards with a speed of 18 m/s. What movement will he make in 3 seconds?

2. An arrow fired from a bow vertically upwards at a speed of 25 m/s hits the target after 2 s. What was the speed of the arrow when it hit the target?

3. A ball was fired vertically upwards from a spring pistol, which rose to a height of 4.9 m. With what speed did the ball fly out of the pistol?

4. The boy threw the ball vertically upwards and caught it after 2 s. What is the height of the ball and what is its initial speed?

5. With what initial speed should the body be thrown vertically upwards so that after 10 s it moves downwards at a speed of 20 m/s?

6. “Humpty Dumpty was sitting on a wall (20 m high),

Humpty Dumpty collapsed in his sleep.

Do you need all the royal cavalry, all the royal army,

to Humpty, to Humpty, Humpty Dumpty,

Dumpty-Humpty collect "

(if it crashes only at 23 m/s?)

So is all the royal cavalry needed?

7. Now the thunder of sabers, spurs, sultan,
And the chamber junker caftan
Patterned - seductive beauties,
Was it not a temptation
When from the guard, others from the court
Came here on time!
Women shouted: hurrah!
And they threw caps into the air.

"Woe from Wit".

The girl Ekaterina threw her bonnet up at a speed of 10 m/s. At the same time, she stood on the balcony of the 2nd floor (at a height of 5 meters). How long will the cap be in flight if it falls under the feet of the brave hussar Nikita Petrovich (naturally standing under the balcony on the street).

1588. How to determine the acceleration of free fall, having at its disposal a stopwatch, a steel ball and a scale up to 3 m high?

1589. What is the depth of the shaft if a stone freely falling into it reaches the bottom 2 s after the fall begins.

1590. The height of the Ostankino television tower is 532 m. A brick was dropped from its highest point. How long will it take him to hit the ground? Air resistance is ignored.

1591. The building of the Moscow state university on Sparrow Hills has a height of 240 m. A piece of facing has come off from the upper part of its spire and is falling freely down. How long will it take to reach the ground? Air resistance is ignored.

1592. A stone falls freely from a cliff. What distance will it cover in the eighth second from the beginning of the fall?

1593. A brick falls freely from the roof of a building 122.5 m high. What distance will the brick travel in the last second of its fall?

1594. Determine the depth of the well if the stone that fell into it touched the bottom of the well after 1 s.

1595. A pencil falls from a table 80 cm high to the floor. Determine the fall time.

1596. A body falls from a height of 30 m. What distance does it travel during the last second of its fall?

1597. Two bodies fall from different heights, but reach the ground at the same time; in this case, the first body falls for 1 s, and the second - for 2 s. How far from the ground was the second body when the first began to fall?

1598. Prove that the time during which a body moving vertically upwards reaches greatest height h is equal to the time during which the body falls from this height.

1599. A body moves vertically downwards with an initial velocity. What are the simplest movements that can be decomposed into such a movement of the body? Write formulas for the speed and distance traveled for this movement.

1600. A body is thrown vertically upward at a speed of 40 m/s. Calculate at what height the body will be after 2 s, 6 s, 8 s and 9 s, counting from the beginning of the movement. Explain answers. To simplify the calculations, take g equal to 10 m/s2.

1601. With what speed must a body be thrown vertically upwards so that it comes back in 10 s?

1602. An arrow is launched vertically upwards with an initial speed of 40 m/s. In how many seconds will it fall back to the ground? To simplify the calculations, take g equal to 10 m/s2.

1603. The balloon rises vertically upwards uniformly at a speed of 4 m/s. A load is suspended from a rope. At an altitude of 217 m, the rope breaks. How many seconds will it take for the weight to hit the ground? Take g equal to 10 m/s2.

1604. A stone is thrown vertically upwards with an initial speed of 30 m/s. 3 s after the start of the movement of the first stone, the second stone was also thrown upwards with an initial speed of 45 m/s. At what height will the stones meet? Take g = 10 m/s2. Ignore air resistance.

1605. A cyclist climbs up a slope 100 m long. The speed at the beginning of the ascent is 18 km / h, and at the end 3 m / s. Assuming the movement is uniformly slow, determine how long the ascent took.

1606. Sledges move down the mountain with uniform acceleration with an acceleration of 0.8 m/s2. The length of the mountain is 40 m. Having rolled down the mountain, the sled continues to move uniformly and stops after 8 s ....

This video tutorial is for self-study topic "Movement of a body thrown vertically upwards". During this lesson, students will gain an understanding of the motion of a body in free fall. The teacher will talk about the movement of a body thrown vertically upwards.

In the previous lesson, we considered the issue of the motion of a body that was in free fall. Recall that we call free fall (Fig. 1) such a movement that occurs under the action of gravity. The force of gravity is directed vertically downward along the radius towards the center of the Earth, acceleration of gravity while equal to .

Rice. 1. Free fall

How will the movement of a body thrown vertically upwards differ? It will differ in that the initial velocity will be directed vertically upwards, i.e., it can also be considered along the radius, but not towards the center of the Earth, but, on the contrary, from the center of the Earth upwards (Fig. 2). But the acceleration of free fall, as you know, is directed vertically downwards. So, we can say the following: the movement of the body vertically upwards in the first part of the path will be slow motion, and this slow motion will also occur with free fall acceleration and also under the action of gravity.

Rice. 2 Movement of a body thrown vertically upwards

Let's turn to the figure and see how the vectors are directed and how it fits with the frame of reference.

Rice. 3. Movement of a body thrown vertically upwards

In this case, the reference system is connected to the earth. Axis Oy is directed vertically upwards, as is the initial velocity vector. The downward force of gravity acts on the body, which imparts to the body the acceleration of free fall, which will also be directed downward.

The following thing may be noted: the body will move slow, will rise to a certain height, and then will start rapidly fall down.

We have designated the maximum height, while .

The movement of a body thrown vertically upwards occurs near the surface of the Earth, when the free fall acceleration can be considered constant (Fig. 4).

Rice. 4. Near the surface of the Earth

Let us turn to the equations that make it possible to determine the speed, instantaneous speed and distance traveled during the considered movement. The first equation is the velocity equation: . The second equation is the equation of motion for uniformly accelerated motion: .

Rice. 5. Axis Oy pointing up

Consider the first frame of reference - the frame of reference associated with the Earth, the axis Oy directed vertically upwards (Fig. 5). The initial velocity is also directed vertically upwards. In the previous lesson, we already said that the acceleration of free fall is directed downward along the radius towards the center of the Earth. So, if we now reduce the velocity equation to a given reference frame, then we get the following: .

It is a projection of the speed at a certain point in time. The equation of motion in this case is: .

Rice. 6. Axis Oy pointing down

Consider another reference system, when the axis Oy directed vertically downwards (Fig. 6). What will change from this?

. The projection of the initial velocity will be with a minus sign, since its vector is directed upwards, and the axis of the selected reference system is directed downwards. In this case, the acceleration of free fall will be with a plus sign, because it is directed downwards. Motion equation: .

Another very important concept to consider is the concept of weightlessness.

Definition.Weightlessness- a state in which the body moves only under the influence of gravity.

Definition. Weight- the force with which the body acts on the support or suspension due to attraction to the Earth.

Rice. 7 Illustration for weight determination

If a body near the Earth or at a short distance from the Earth's surface moves only under the action of gravity, then it will not act on the support or suspension. This state is called weightlessness. Very often, weightlessness is confused with the concept of the absence of gravity. In this case, it must be remembered that weight is the action on the support, and weightlessness- this is when there is no effect on the support. Gravity is a force that always acts near the surface of the Earth. This force is the result of gravitational interaction with the Earth.

Let's take a look at one more important point associated with the free fall of bodies and the movement vertically upwards. When the body moves up and moves with acceleration (Fig. 8), an action occurs, leading to the fact that the force with which the body acts on the support exceeds the force of gravity. If this happens, this state of the body is called overload, or the body itself is said to be overloaded.

Rice. 8. Overload

Conclusion

The state of weightlessness, the state of overload - these are extreme cases. Basically, when a body is moving on a horizontal surface, the weight of the body and the force of gravity most often remain equal to each other.

Bibliography

Kikoin I.K., Kikoin A.K. Physics: Proc. for 9 cells. avg. school - M.: Enlightenment, 1992. - 191 p.
Sivukhin D.V. General course physics. - M .: State publishing house of technical
theoretical literature, 2005. - T. 1. Mechanics. - S. 372.
Sokolovich Yu.A., Bogdanova G.S. Physics: Handbook with examples of problem solving. - 2nd edition, redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.

Internet portal "eduspb.com" ()
Internet portal "physbook.ru" ()
Internet portal "phscs.ru" ()

Homework

Let the body begin to fall freely from rest. In this case, the formulas of uniformly accelerated motion without initial velocity with acceleration are applicable to its movement. Let us denote the initial height of the body above the ground through, the time of its free fall from this height to the ground - through and the speed reached by the body at the moment of falling to the ground - through. According to the formulas of § 22, these quantities will be related by the relations

(54.1)

(54.2)

Depending on the nature of the problem, it is convenient to use one or the other of these relations.

Let us now consider the motion of a body, which is given some initial velocity , directed vertically upwards. In this problem, it is convenient to assume that the upward direction is positive. Since the acceleration of free fall is directed downwards, the motion will be uniformly slowed down with negative acceleration and with a positive initial velocity. The speed of this movement at a moment of time is expressed by the formula

and the height of the lift at this moment above the starting point - the formula

(54.5)

When the speed of the body decreases to zero, the body will reach highest point lifting; it will happen at the moment for which

After this moment, the speed will become negative and the body will start to fall down. So, the time of lifting the body

Substituting the rise time into formula (54.5), we find the height of the body rise:

(54.8)

The further movement of the body can be considered as a fall without initial velocity (the case considered at the beginning of this section) from a height. Substituting this height into formula (54.3), we find that the speed that the body reaches at the moment it falls to the ground, i.e., returning to the point from which it was thrown upwards, will be equal to the initial speed of the body (but, of course, will be directed oppositely - down). Finally, from formula (54.2) we conclude that the time the body falls from the highest point is equal to the time the body rises to this point.

5 4.1. A body falls freely without initial speed from a height of 20 m. At what height will it reach a speed equal to half the speed at the moment of falling to the ground?

54.2. Show that a body thrown vertically upward passes each point of its trajectory with the same modulo speed on the way up and on the way down.

54.3. Find the speed when a stone thrown from a tower of height hits the ground: a) without initial speed; b) with initial speed directed vertically upwards; c) with initial speed directed vertically downwards.

54.4. A stone thrown vertically upwards passed the window 1 s after the throw on the way up and 3 s after the throw on the way down. Find the height of the window above the ground and the initial speed of the stone.

54.5. When firing vertically at air targets, a projectile fired from an anti-aircraft gun reached only half the distance to the target. A projectile fired from another gun hit its target. How many times greater is the initial velocity of the projectile of the second gun than the velocity of the first?

54.6. What is the maximum height to which a stone thrown vertically upwards will rise if, after 1.5 s, its speed has halved?

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vertical movement of the body. The motion of a body thrown vertically upwards

The essence of the movement vertically upwards

How to solve problems

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