All bodies in airless space fall with the same acceleration. But why is this happening? Why is the acceleration of a freely falling body independent of its mass? To answer these questions, we have to think carefully about the meaning of the word "mass."

Let us dwell first of all on the course of Galileo's reasoning, with which he tried to prove that all bodies must fall with the same acceleration. Do we not come, arguing with similar images, for example, to the conclusion that in electric field do all the charges 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 connect these charges. How should the composite charge move now: faster or slower than a large charge? One thing is certain that the force acting on the compound charge from the electric field will be greater than the forces experienced by each charge separately. However, this information is still insufficient to determine the acceleration of the body; you also need to know the total mass of the compound charge. For lack of data, we must interrupt our discussion of the motion of a compound 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 movement of mass in a gravitational field and the movement 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 side of the electric field, the second determines the acceleration at 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. Galileo had enough of one value because he considered the gravitational charge 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 at a given force is called "inert mass." So, for example, the mass referred to in the special theory of relativity is inertial mass.

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

What do we mean, for example, by the statement that a loaf of bread weighs 1 kg? This is the bread that the Earth attracts to itself with power at 1 Kg (of course, bread 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 second loaf has more mass (more precisely, twice as much) than the first.

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

AT everyday life by weight we mean 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 manages to visit other planets, he will have the opportunity to directly verify that body weight depends on the mass relative to which he is measured. Imagine that astronauts, going to Mars, took with them a loaf of bread that weighs 1 kg. Weighing it on the surface of Mars, they find that the weight of the loaf was equal to 380 r... The heavy weight of the bread did not change during the flight, but the weight of the bread decreased by almost three times. 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 be saturated in exactly the same way, no matter where it is - on Earth or on Mars. From this example it is clear that the 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 thanks to 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 squad. Here you can't stop the train right away - you have to slow down longer. Why does it take different time to stop trains? Usually the answer is that the second train was heavier than the first - this is the reason. This answer is inaccurate. What does a steam locomotive driver care about the weight of the train? It is only important to him what resistance the train has to reduce speed. Why should we assume that the train, which the Earth attracts to itself more strongly, is more resistant to speed changes? True, everyday observations show that this is so, but it may turn out that this is pure chance. 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 considerable time for this. We must look for the reason in something else. 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", that is, it changes its speed more slowly, then they say that it has a large inertia. We have seen that a train with a lower mass has less inertia than a train with a higher mass. Here we again used the word "mass", but in a different sense. Above, mass characterizes the attraction of a body by other bodies, but here it characterizes the inertia of a body. Therefore, in order to eliminate the confusion in the use of the same word "mass" in two different meanings, they say "heavy mass" and "inert mass". While heavy mass characterizes the gravitational effect on a body from other bodies, inertial mass characterizes the body's inertia. If the heavy weight of the body doubles, then the force of attraction of its other bodies will double. If the inert mass doubles, then the acceleration acquired by the body under the action of this force will halve. If, with an inert mass that is twice as large, it is required that the acceleration of the body remains the same, then it will need to apply twice as much force to it.

What happened if all bodies had inert mass equal to heavy mass? Suppose we have, for example, a piece of iron and a stone, and the inert mass of a piece of iron is three times the inert mass of a stone. This means that in order to impart the same accelerations to these bodies, a piece of iron needs to be acted upon three times more force than a stone. Suppose now 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 greater than the heavy mass of a stone; a piece of iron will be attracted by the Earth three times more than a stone. But for the transmission of equal accelerations, it is precisely three times the force that is required. Therefore, a piece of iron and a stone will fall to the Earth with equal accelerations.

From the foregoing it follows that with the equality of inert and heavy masses, 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 all bodies have an inert mass equal to their heavy mass.

Inertial mass and heavy mass are different concepts that are not logically related to each other. Each of them characterizes a certain property of the body. And if experience shows that the inert and heavy masses are equal, then this means that in fact we, using two different concepts, have characterized the same property of the body. The body has only one mass. The fact that we previously attributed masses of two genera to him was due only to our insufficient knowledge of nature. Right now, we can rightfully say that heavy body mass is equivalent to inert mass. Consequently, the ratio of heavy and inert mass is to some extent analogous to the ratio of mass (more precisely, inert 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 inert and heavy masses. Since this equality has been established empirically, here one must certainly reckon with errors that inevitably appear in all measurements. According to Newton's estimate, for a body with a heavy mass at 1 Kg the inert mass may differ from the kilogram by no more than 1 g.

The German astronomer Bessel used a pendulum to study the ratio of inert 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, precise measurements carried out with various bodies, including living beings, showed that there is no such dependence. Heavy mass equals inert mass. Given the accuracy of his experience, Bessel could argue that an inert body mass of 1 kg can differ from the heavy mass by no more than 0.017 g. In 1894 the Hungarian physicist R. Eötvös succeeded in comparing inert and heavy masses with very high accuracy. From the measurements it followed that the inert body mass at 1 Kg may differ from heavy weight by no more than 0.005 mr . Modern measurements have made it possible to reduce the possible error by about a hundred times. Such an accuracy of measurements makes it possible to assert that the inert and heavy masses are really equal.

Particularly interesting experiments were carried out in 1918 by the Dutch physicist Zeeman, who studied the ratio of heavy and inert mass for the radioactive isotope of uranium. Uranium nuclei are unstable and turn into lead and helium nuclei over time. In this case, energy is released in the process of radioactive decay. An approximate estimate shows that under the transformation 1 r pure uranium into lead and helium should be liberated 0.0001 r energy (we saw above that energy can be measured in grams). Hence, we can say that 1 r uranium contains 0.9999 r inert mass and 0.0001 r 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. Such a 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 is enough to say simply that the inert mass of a piece of uranium is equal to 1 g. So is its heavy mass. In radioactive bodies, inert and heavy masses are also equal to each other. Equality of inert and heavy mass is a common property of all bodies of nature.

For example, particle accelerators, imparting energy to particles, thereby increase their weight. If, for example, the 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, sometimes powerful electron accelerators are called electron weights).

Free fall is the movement of objects vertically downward or vertically upward. This is a uniformly accelerated movement, but its special kind. For this motion, all formulas and laws of uniformly accelerated motion are valid.

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

Free fall acceleration is constant.
This means whatever body is flying up or down, its speed will change in the same way. BUT with one caveat, if the air resistance force can be neglected.

The acceleration of gravity is usually denoted by a letter other than acceleration. But free fall acceleration and acceleration are one and the same physical quantity and they have the same physical meaning. They participate in the same way in 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 simultaneously. Whatever one may say, whether they are in a vacuum or in air, their mass is different. The answer is simple. The force that makes the bodies fall (gravity), caused by the gravitational field of the Earth, these bodies are different. For a stone, it is larger (since a stone has more mass), for a feather it is less. But there is no dependence here: the greater the force, the greater the acceleration! Let's compare, we act with equal force on a heavy cabinet and a light bedside table. Under the influence of this force, the bedside table will move faster. And in order for the cabinet and the bedside table to move in the same way, the cabinet must be influenced more strongly than the bedside table. The Earth is doing the same. It attracts heavier bodies with greater 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.

Let us consider separately the question of the resulting air resistance. Take two identical sheets of paper. We crumple one of them and simultaneously 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 due to air resistance.

Consider a body falling from a certain height h without initial speed. If the coordinate axis OU is directed upward, aligning the origin of coordinates with the Earth's surface, we will receive the main characteristics of this movement.

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

IMPORTANT! Since the rise of the body to the 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 will land will be equal to the speed with which it thrown up. In this case, the time the body rises to the maximum height will be equal to the time the body falls from this height to the ground level. Thus, the entire flight time will be twice the rise or fall time. The speed of the body at the same level when lifting and falling will also be the same.

The main thing to remember

1) Direction of acceleration during free fall of the body;
2) The numerical value of the acceleration due to gravity;
3) Formulas

Derive a formula for determining the time a body falls from a certain height h without initial speed.

Derive a formula for determining the time of rise of the body to the maximum height, thrown at the initial speed v0

Derive a formula for determining the maximum lifting height of a body thrown vertically upward with an initial speed v0

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

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

If the fall occurs in air, then in addition to the acceleration of free fall, one more appears; it is directed against the movement of the body (if the body simply 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 (the speed and shape of a body, for example), but the acceleration that this force will give to a body depends on the mass of this body (Newton's second law - F \u003d ma, where a is acceleration). That is, if conventionally, the bodies "fall" with the same acceleration, but to different degrees "slow down" under the action of the resistance force of the medium. In other words, the foam ball will "slow down" more actively against the air as long as its mass is less than that of a nearby flying lead. In a vacuum, there is no resistance 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 reservation. 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 a feather. The fact is that BOTH masses appear in the Law of Gravitation, which describes the force of pairwise attraction of massive bodies. 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 \u200b\u200bfor the pellet and the feather will be vanishingly small. If, for example, we are talking about the "fall" of two balls in half and a quarter of the Earth's mass, 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 to 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 this moment approaches the stone at an insignificantly (vanishing) small distance. No comment. Just think about it before bed.

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Free fall is an interesting, but at the same time, rather difficult question, since all listeners are surprised and distrustful of the fact that all bodies, regardless of their mass, fall with the same acceleration and even with equal speeds, if there is no resistance of the environment. 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 mathematical calculations and the concept of a directly proportional relationship between gravity and mass are not enough. We need more convincing images than reasoning according to the formula g \u003d Ftyazh / m that when the mass doubles, the force of gravity also doubles and the twos are reduced (that is, as a result, the formula takes on the same form). Then analogous conclusions are made for a three, a four, etc. But the students do not see a real explanation behind the formulas. The formula remains, as it were, by itself, and life experience prevents one from agreeing with the teacher's story. And no matter how much the teacher speaks, does not persuade, but there will be no solid knowledge, logically grounded, 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 enough simple experiments proving the influence of air on the movement of a body in any environment and amusing theoretical reasoning, which, on the one hand, can interest many with their clarity, and on the other hand, will allow you to quickly and efficiently assimilate the material under study.

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 is made with the use of 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. Title
2. Definition of the term "Free fall"
3. Portrait of Galileo
4. Galileo's experiments. 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: Ftyaz \u003d 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 direct and inverse proportional dependence, these circles, at the click of the mouse, simultaneously begin to increase or decrease in the same number of times.
6. Gravity is proportional to mass. But this time it's shown mathematically. Animation allows you to substitute the same factors in both the numerator and the denominator of the formula for the acceleration of gravity. These numbers are reduced (which is also shown in the animation) and the formula remains 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 due to gravity 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 don't count the air and fly to the ground,
The speed is growing, it's already clear to me.
Everything is the same every second:
The Earth will help us all to add "ten".
I increase the speed by meters per second.
As soon as I reach the ground, maybe I'll calm down.
I'm glad that I have time, knowing the acceleration,
Experience free fall.
But I guess it's better next time
I'll climb the mountains, maybe the Caucasus:
"G" will be less there. Only here's the trouble
You step down and again the numbers, as always,
They will run at a gallop - do not stop.
At least, actually, the air will slow down.
No. Let's go to the Moon or Mars.
It is safer to experience there many times over.
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 the air and in airless space (animation).

1. The slide shows a setup for demonstrating the experience of moving bodies in an airless space. Newton's tube is connected with a hose to the Komovsky pump. After a sufficient vacuum has been created in the tube, the bodies in it (pellet, cork and feather) fall almost simultaneously.
2. Animation: "Falling bodies in Newton's tube." Bodies: fraction, coin, cork, feather.
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 (green vector) appears on the screen. For the second body (plate) with a larger surface area, the air resistance is greater, and the resultant of gravity and air resistance is less than for a ball.
4. 
   We take two paper sheets the same mass... One of them was crumpled up. Sheets fall from different speeds and accelerations. This is how we prove that two bodies of equal mass, having different shapes, fall in the air at different speeds.
5. Photos of experiments without Newton's tube showing the role of air in resisting the motion of bodies.
   We take a textbook and a paper sheet, the length and width of which is less than that of the book. The masses of these two bodies are naturally different, but they will fall from the same speeds and accelerations, if we remove the influence of air resistance for the sheet, that is, put the sheet on the book. If the bodies are raised above the surface of the earth 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, we can offer another example. When we see that two bodies with masses m and 2m, having an initial velocity of zero and the same acceleration, require the application of forces that are also 2 times different, we are not surprised. This is during normal movement on a horizontal surface. But the same task and the same reasoning in relation to falling bodies already seem incomprehensible.
7. For an analogy, we need to rotate the horizontal drawing by 900 and compare it with the 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 for a body of 2m 2 horses are needed in order for the second body to keep up with the first and move with the same acceleration. But there will be similar explanations for the vertical movement. Only we will talk about the influence of the Earth. The force of gravity acting on a body with a mass of 2m is 2 times greater than for the first body with a mass of 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 less, then the more massive body would not keep up with the smaller body. It's the same as looking at horse racing in the previous slide. Thus, while studying the topic of the free fall of bodies, we do not seem to think about the fact that without the influence of the Earth, these bodies would have to "hang" in space in place. No one would change their speed equal to zero. We are simply too accustomed to gravity and no longer notice its role. Therefore, it seems to us so strange the statement about the equality of the acceleration of gravity for bodies of very different masses.