And one more important condition - in a vacuum. And not speed, but acceleration in this case. Yes, to a certain degree of approximation it is so. Let's figure it out.

So, if two bodies fall from the same height in a vacuum, they will fall at the same time. Even Galileo Galilei once experimentally proved that bodies fall to the Earth (namely with a capital letter - we are talking about a planet) with the same acceleration, regardless of their shape and mass. The legend says that he took a transparent tube, put a pellet and a feather in it, but pumped out the air from there. And it turned out that being in such a tube, both bodies fell down at the same time. The fact is that every body in the Earth's gravitational field experiences the same acceleration (on average g ~ 9.8 m / s²) of free fall, regardless of its mass (in fact, this is not entirely true, but in the first approximation - Yes, in fact, in physics this is not uncommon - read to the end).

If the fall occurs in the air, then in addition to the acceleration of free fall, another one arises; it is directed against the movement of the body (if the body just falls, then against the direction of free fall) and is caused by the force of air resistance. The force itself depends on a bunch of factors (speed and shape of the body, for example), but the acceleration that this force will give to the body depends on the mass of this body (Newton's second law is F = ma, where a is acceleration). That is, if conditionally, then the bodies "fall" with the same acceleration, but "slow down" to a different extent under the influence of the resistance force of the medium. In other words, the foam ball will more actively "slow down" on the air as long as its mass is less than that of a nearby lead ball. There is no resistance in a vacuum, and both balls will fall approximately (up to the depth of the vacuum and the accuracy of the experiment) at the same time.

Well, in conclusion, the promised disclaimer. In the tube mentioned above, the same as that of Galileo, even under ideal conditions, the pellet will fall a negligible number of nanoseconds earlier, again due to the fact that its mass is negligible (compared to the mass of the Earth) differs from the mass of the feather. The fact is that in the Law of universal gravitation, which describes the force of pairwise attraction of massive bodies, BOTH masses appear. That is, for each pair of such bodies, the resulting force (and hence the acceleration) will depend on the mass of the "falling" body. However, the contribution of the pellet to this force will be negligible, which means that the difference between the acceleration values ​​for the pellet and the feather will be vanishingly small. If, for example, we are talking about the "fall" of two balls at half and a quarter of the mass of the Earth, respectively, then the first one will "fall" noticeably earlier than the second. The truth about the "fall" is difficult to speak here - such a mass will noticeably displace the Earth itself.

By the way, when a pellet or, say, a stone falls on the Earth, then, according to the same Law of universal gravitation, not only the stone overcomes the distance to the Earth, but the Earth at that moment approaches the stone at an negligibly (vanishingly) small distance. No comment. Just think about it before bed.

Free fall is the movement of objects vertically downwards or vertically upwards. This is a uniformly accelerated movement, but a special kind of it. All formulas and laws of uniformly accelerated motion are valid for this motion.

If the body flies vertically downwards, then it accelerates, in this case the velocity vector (directed vertically downwards) coincides with the acceleration vector. If the body flies vertically upwards, then it slows down, in this case the velocity vector (directed upwards) does not coincide with the direction of acceleration. The acceleration vector in free fall is always directed vertically downwards.

Acceleration in the free fall of bodies is a constant value.
This means that no matter what body is flying up or down, its speed will change the same way. BUT with one caveat, if the force of air resistance can be neglected.

Free fall acceleration is usually denoted by a letter different from acceleration. But free fall acceleration and acceleration are one and the same physical quantity and they have the same physical meaning. They participate equally in the formulas for uniformly accelerated motion.

We write the "+" sign in the formulas when the body flies down (accelerates), the "-" sign - when the body flies up (slows down)

Everyone knows from school physics textbooks that in a vacuum a pebble and a feather fly the same way. But few people understand why, in a vacuum, bodies of different masses land at the same time. Like it or not, whether they are in a vacuum or in air, their mass is different. The answer is simple. The force that causes bodies to fall (gravity) caused by the Earth's gravitational field is different for these bodies. It is larger for a stone (since the stone has more mass), for a feather it is smaller. But there is no dependence here: the greater the force, the greater the acceleration! Let's compare, we act with the same force on a heavy cabinet and a light bedside table. Under the influence of this force, the nightstand will move faster. And in order for the cabinet and bedside table to move in the same way, it is necessary to act on the cabinet more strongly than on the bedside table. The Earth does the same. It attracts heavier bodies with more force than light ones. And these forces are so distributed among the masses that as a result they all fall in a vacuum at the same time, regardless of the mass.

Separately, consider the issue of emerging air resistance. Take two identical sheets of paper. We crumple one of them and at the same time release it from our hands. The crumpled leaf will fall to the ground earlier. Here, the different fall times are not related to body mass and gravity, but are due to air resistance.

Consider a body falling from a certain height h no initial speed. If the OS coordinate axis is directed upwards, aligning the origin of coordinates with the Earth's surface, we obtain the main characteristics of this movement.

A body thrown vertically upwards moves uniformly with the acceleration of free fall. In this case, the velocity and acceleration vectors are directed in opposite directions, and the velocity modulus decreases with time.

IMPORTANT! Since the rise of the body to its maximum height and the subsequent fall to the ground level are absolutely symmetrical movements (with the same acceleration, just one slowed down and the other accelerated), the speed with which the body lands will be equal to the speed with which it tossed up. In this case, the time for the body to rise to the maximum height will be equal to the time for the body to fall from this height to the ground level. Thus, the entire flight time will be double the time of ascent or fall. The speed of the body at the same level during the ascent and during the fall will also be the same.

The main thing to remember

1) The direction of acceleration in the free fall of the body;
2) Numerical value of free fall acceleration;
3) Formulas

Derive a formula for determining the time of a body falling from a certain height h no initial speed.

Derive a formula for determining the time it takes a body to rise to its maximum height, thrown with an initial speed v0

Derive a formula for determining the maximum height of a body thrown vertically upwards with an initial velocity v0

All bodies in airless space fall with the same acceleration. But why is this happening? Why does the acceleration of a freely falling body not depend on its mass? To answer these questions, we will have to think carefully about the meaning of the word "mass".

Let us first of all dwell on the course of Galileo's reasoning, with which he tried to prove that all bodies must fall with the same acceleration. Shall we not come, reasoning with such images, for example, to the conclusion that in an electric field all charges also move with the same acceleration?

Let there be two electric charges - large and small; suppose that in a given electric field, a large charge moves faster. Let's combine these charges. How should the composite charge now move: faster or slower than the large charge? One thing is certain, that the force acting on the composite charge from the electric field will be greater than the forces experienced by each charge separately. However, this information is still not enough to determine the acceleration of the body; you also need to know the total mass of the composite charge. Due to lack of data, we must interrupt our discussion of the motion of a composite charge.

But why didn't Galileo encounter similar difficulties when he discussed the fall of heavy and light bodies? What is the difference between the motion of a mass in a gravitational field and the motion of a charge in an electric field? It turns out that there is no fundamental difference here. To determine the movement of a charge in an electric field, we must know the magnitude of the charge and mass: the first of them determines the force acting on the charge from the electric field, the second determines the acceleration for a given force. To determine the motion of a body in a gravitational field, two quantities must also be taken into account: the gravitational charge and its mass. The gravitational charge determines the magnitude of the force with which the gravitational field acts on the body, while the mass determines the acceleration of the body in the case of a given force. One quantity was enough for Galileo because he considered the gravitational charge to be equal to the mass.

Usually physicists do not use the term "gravitational charge", but instead say "heavy mass". To avoid confusion, the mass that determines the acceleration of a body for a given force is called "inertial mass". Thus, for example, the mass referred to in the special theory of relativity is an inertial mass.

Let us characterize the heavy and inertial masses somewhat more precisely.

What do we understand, for example, by the statement that a loaf of bread weighs 1 kg? This is the bread that the Earth draws to itself with force. in 1 kg (of course, bread also attracts the Earth with the same force). Why does the Earth attract one loaf with a force of 1 kg, and another, large, say, with a force of 2 kg? Because the second loaf has more bread than the first. Or, as they say, the mass of the second loaf is greater (more precisely, twice as much) than the first.

Each body has a certain weight, and the weight depends on the heavy mass. A heavy mass is a characteristic of a body that determines its weight, or, in other words, a heavy mass determines the magnitude of the force with which the body in question is attracted by other bodies. Thus, the quantities t and M, appearing in formula (10) are heavy masses. It must be borne in mind that a heavy mass is a certain quantity that characterizes the amount of matter contained in a body. Body weight, on the contrary, depends on external conditions.

In everyday life, by weight we understand the force with which the body is attracted by the Earth, we measure the weight of the body in relation to the Earth. We could just as well talk about the weight of a body relative to the Moon, the Sun, or any other body. When a person succeeds in visiting other planets, he will be able to directly verify that the weight of a body depends on the mass relative to which it is measured. Imagine that the astronauts, going to Mars, took with them a loaf of bread, which weighs 1 kg. When they weigh it on the surface of Mars, they find that the loaf weighs 380 G. The heavy mass of bread did not change during the flight, but the weight of the bread almost halved. The reason is clear: the heavy mass of Mars is less than the heavy mass of the Earth, so the attraction of bread on Mars is less than on Earth. But this bread will saturate in exactly the same way, regardless of where it is - on Earth or on Mars. This example shows that a body must be characterized not by its weight, but by its heavy mass. Our system of units is chosen in such a way that the weight of the body (in relation to the Earth) is numerically equal to the heavy mass, only because of this we do not need to distinguish between heavy mass and body weight in everyday life.

Consider the following example. Let a short freight train arrive at the station. The brakes are applied and the train stops immediately. Then comes the heavyweight. Here you can’t stop the train right away - you have to slow down longer. Why does it take different times for trains to stop? Usually they answer that the second train was heavier than the first - this is the reason. This answer is inaccurate. What does the engine driver care about the weight of the train? The only thing that matters to him is what kind of resistance the train provides to a decrease in speed. Why should we assume that the train, which the Earth pulls towards itself more strongly, resists the change in speed more stubbornly? True, everyday observations show that this is the case, but it may turn out that this is pure coincidence. There is no logical connection between the weight of the train and the resistance it provides to the change in speed.

So, we cannot explain by the weight of the body (and, consequently, by the heavy mass) the fact that under the action of the same forces one body obediently changes its speed, while the other requires a considerable time for this. We must look for the cause elsewhere. The property of a body to resist a change in speed is called inertia. Earlier we have already noted that in Latin "inertia" means laziness, lethargy. If the body is "lazy", i.e., changes its speed more slowly, then they say that it has a large inertia. We have seen that a train with a smaller mass has less inertia than a train with a larger mass. Here we again used the word "mass", but in a different sense. Above, the mass characterized the attraction of the body by other bodies, but here it characterizes the inertia of the body. That is why, in order to eliminate confusion in the use of the same word "mass" in two different meanings, they say "heavy mass" and "inert mass". While a heavy mass characterizes the gravitational effect on a body from other bodies, an inertial mass characterizes the body's inertia. If the heavy mass of a body doubles, then the force of attraction by its other bodies will double. If the inertial mass is doubled, then the acceleration acquired by the body under the action of this force will be halved. If, with an inertial mass twice as large, it is required that the acceleration of the body remain the same, then twice as much force will need to be applied to it.

What would happen if the inertial mass of all bodies was equal to the heavy mass? Suppose we have, for example, a piece of iron and a stone, and the inertial mass of the piece of iron is three times greater than the inertial mass of the stone. This means that in order to impart the same accelerations to these bodies, a piece of iron must be acted upon by three times more force than a stone. Let us now assume that the inertial mass is always equal to the heavy one. This means that the heavy mass of a piece of iron will be three times the heavy mass of stone; a piece of iron will be attracted to the Earth three times stronger than a stone. But to communicate equal accelerations, exactly three times as much force is required. Therefore, a piece of iron and a stone will fall to the Earth with equal accelerations.

It follows from the foregoing that if the inertial and heavy masses are equal, all bodies will fall to the Earth with the same acceleration. Experience really shows that the acceleration of all bodies in free fall is the same. From this we can conclude that for all bodies the inertial mass is equal to the heavy mass.

Inertial mass and heavy mass are different concepts, logically unrelated. Each of them characterizes a certain property of the body. And if experience shows that the inertial and heavy masses are equal, then this means that in fact we have characterized the same property of the body with the help of two different concepts. The body has only one mass. The fact that we used to attribute two kinds of masses to it was due only to our insufficient knowledge of nature. With full right at the present time we can say that the heavy mass of the body is equivalent to the inertial mass. Consequently, the ratio of heavy and inertial mass is to some extent analogous to the ratio of mass (more precisely, inertial mass) and energy.

Newton was the first to show that the laws of free fall discovered by Galileo take place due to the equality of inertial and heavy masses. Since this equality has been established empirically, here one must necessarily reckon with errors that inevitably appear in all measurements. According to Newton's estimate, for a body with a heavy mass in 1 kg the inertial mass can differ from the kilogram by no more than 1 g.

The German astronomer Bessel used a pendulum to study the relationship between inertial and heavy mass. It can be shown that if the inertial mass of the bodies is not equal to the heavy mass, the period of small oscillations of the pendulum will depend on its weight. Meanwhile, accurate measurements carried out with various bodies, including living beings, showed that there is no such dependence. Heavy mass equals inertial mass. Given the accuracy of his experience, Bessel could argue that the inertial mass of a body in 1 kg may differ from the heavy mass by no more than 0.017 g. In 1894, the Hungarian physicist R. Eötvös managed to compare the inertial and heavy masses with very high accuracy. It followed from the measurements that the inertial mass of the body in 1 kg may differ from the heavy mass by no more than 0.005 mG . Modern measurements have made it possible to reduce the possible error by about a hundred times. Such measurement accuracy makes it possible to assert that the inertial and heavy masses are indeed equal.

Particularly interesting experiments were carried out in 1918 by the Dutch physicist Zeeman, who studied the ratio of heavy and inertial mass for the radioactive isotope of uranium. Uranium nuclei are unstable and eventually turn into lead and helium nuclei. In the process of radioactive decay, energy is released. An approximate estimate shows that, upon transformation 1 G pure uranium into lead and helium should be released 0.0001 G energy (we saw above that energy can be measured in grams). So we can say that 1 G uranium contains 0.9999 G inertial mass and 0.0001 G energy. Zeeman's measurements showed that the heavy mass of such a piece of uranium is 1 g. This means that 0.0001 g of energy is attracted by the Earth with a force of 0.0001 g. This result was to be expected. We have already noted above that it makes no sense to distinguish between energy and inertial mass, because both of them characterize the same property of the body. Therefore, it suffices to say simply that the inertial mass of a piece of uranium is 1 g. So is its heavy mass. In radioactive bodies, the inert and heavy masses are also equal to each other. The equality of inert and heavy mass is a common property of all bodies of nature.

For example, elementary particle accelerators, imparting energy to particles, thereby increase their weight. If, for example, electrons emitted from the accelerator,. have an energy that is 12,000 times greater than the energy of electrons at rest, then they are 12,000 times heavier than the latter. (For this reason, powerful electron accelerators are sometimes referred to as electron "weighters").

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Free fall is an interesting, but at the same time rather complicated question, since all listeners are surprised and distrusted by the fact that all bodies, regardless of their mass, fall with the same acceleration and even with equal speeds, if there is no environmental resistance. In order to overcome this prejudice, the teacher has to spend a lot of time and effort. Although there are times when a teacher asks a colleague in secret from the students: “Why are the speed and acceleration the same?” That is, it turns out that sometimes the teacher mechanically presents some kind of truth, although at the everyday level he himself remains among the doubters. This means that only mathematical calculations and the concept of a directly proportional relationship between gravity and mass are not enough. We need more convincing images than the arguments according to the formula g \u003d Fheavy / m that when the mass doubles, the force of gravity also doubles and the deuces are reduced (that is, as a result, the formula takes on its former form). Then similar conclusions are made for three, four, etc. But behind the formulas, students do not see a real explanation. The formula remains, as it were, on its own, and life experience makes it difficult to agree with the teacher's story. And no matter how much the teacher says, does not convince, but there will be no solid knowledge, logically justified, leaving a deep mark in the memory. Therefore, as experience shows, in such a situation, a different approach is needed, namely, the impact on the emotional level - to surprise and explain. In this case, one can do without the cumbersome experiment with Newton's tube. Quite simply, simple experiments proving the influence of air on the movement of a body in any medium and amusing theoretical discussions are quite enough, which, on the one hand, with their clarity, can interest many, and on the other hand, will allow you to quickly and efficiently assimilate the material being studied.

The presentation on this topic contains slides corresponding to the paragraph “Free fall of bodies” studied in grade 9, and also reflects the above problems. Let's consider the content of the presentation in more detail, since it was made using animation and, therefore, it is necessary to explain the meaning and purpose of individual slides. The description of the slides will be in accordance with their numbering in the presentation.

1. header
2. Definition of Free Fall
3. Portrait of Galileo
4. Experiences of Galileo. Two balls of different masses fall from the Leaning Tower of Pisa and reach the ground at the same time. Gravity vectors, respectively, of different lengths.
5. The force of gravity is proportional to the mass: Fgr = mg. In addition to this statement, there are two circles on the slide. One is red, the other is blue, which matches the color of the letters for gravity and mass on this slide. To demonstrate the meaning of the direct and inverse relationship, these circles at the click of the mouse simultaneously begin to increase or decrease by the same number of times.
6. The force of gravity is proportional to the mass. But this time it is shown mathematically. Animation allows you to substitute the same factors in both the numerator and the denominator of the formula for the free fall acceleration. These numbers are reduced (which is also shown in the animation) and the formula becomes the same. That is, here we prove to the students theoretically that in free fall, the acceleration of all bodies, regardless of their mass, is the same.
7. The value of the acceleration of free fall on the surface of the globe is not the same: it decreases from the pole to the equator. But when calculating, we take an approximate value of 9.8 m / s2.
8. 9. Free Fall Poems(after reading them, students should be asked about the content of the poem)

We do not count the air and fly to the ground,
The speed is growing, it's clear to me.
Every second it's the same
To add “ten” to everyone, the Earth will help us.
I add speed in meters per second.
When I reach the ground, maybe I will calm down.
I am glad that I have time, knowing the acceleration,
Experience free fall.
But maybe better next time
I will climb the mountains, maybe the Caucasus:
"g" will be less there. Only here is the trouble
You step down and again the numbers, as always,
Run at a gallop - do not stop.
Although, in general, the air will slow down.
No. Let's go to the Moon or Mars.
The experiments are much safer there.
Less attraction - I learned everything myself,
So, it will be more interesting to jump there.

1. 11. The movement of a light sheet and a heavy ball in air and in airless space (animation).

1. The slide shows an installation for demonstrating experience on the movement of bodies in an airless space. The Newton tube is connected by a hose to the Komovsky pump. After a sufficient vacuum is created in the tube, the bodies in it (shotgun, cork and feather) fall almost simultaneously.
2. Animation: "The fall of bodies in the Newton's tube." Bodies: fraction, coin, cork, pen.
3. Consideration of the resultant forces applied to the body when moving in air. Animation: the force of air resistance (blue vector) is subtracted from the force of gravity (red vector) and the resultant force appears on the screen (green vector). For the second body (plate) with a larger surface area, the air resistance is greater, and the resultant force of gravity and air resistance is less than for a ball.
4. 
   Take two sheets of paper the same mass. One of them crumpled. Leaves fall from different speeds and accelerations. So we prove that two bodies of equal mass, having different shapes, fall in the air at different speeds.
5. Photographs of experiments without Newton's tube, showing the role of air in resisting the movement of bodies.
   We take a textbook and a paper sheet, the length and width of which is less than that of a book. The masses of these two bodies are, of course, different, but they will fall from the same speeds and accelerations, if we remove the influence of air resistance for a sheet, that is, put a sheet on a book. If the bodies are raised above the ground and released separately from each other, then the leaf falls much more slowly.
6. To the question that many do not understand why the acceleration of freely falling bodies is the same and does not depend on the mass of these bodies.
   In addition to the fact that Galileo, considering this problem, proposed replacing one massive body with two of its parts connected by a chain, and analyzing the situation, one more example can be offered. When we see that two bodies with masses m and 2m, having an initial velocity equal to zero and the same acceleration, require the application of forces that also differ by a factor of 2, nothing surprises us. This is during normal movement on a horizontal surface. But the same problem and the same reasoning in relation to falling bodies already seem incomprehensible.
7. For analogy, we need to rotate the horizontal drawing by 900 and compare it with falling bodies. Then it will be seen that there are no fundamental differences. If a body of mass m is pulled by one horse, then 2 horses are needed for a body 2m in order for the second body to keep up with the first and move with the same acceleration. But for vertical movement there will be similar explanations. Only we will talk about the influence of the Earth. The force of gravity acting on a body of mass 2m is 2 times greater than for the first body of mass m. And the fact that one of the forces is 2 times greater does not mean that the body should move faster. This means that if the force were smaller, then the more massive body would not be able to keep up with the smaller body. It's like looking at horse racing on the previous slide. Thus, while studying the subject of the free fall of bodies, we do not seem to think that without the influence of the Earth, these bodies would have to “hang” in space in place. Nobody would change their speed equal to zero. We are simply too accustomed to gravity and no longer notice its role. Therefore, the statement about the equality of the acceleration of free fall for bodies of very different masses seems so strange to us.