Acceleration of gravity. Free Fall of Bodies Why do different bodies fall differently?

Free fall is the movement of objects vertically downwards or vertically upwards. This is uniformly accelerated motion, but its special type. For this motion, all formulas and laws of uniformly accelerated motion are valid.

If a body flies vertically downward, then it accelerates, in this case the velocity vector (directed vertically downward) coincides with the acceleration vector. If a 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 acceleration vector during free fall is always directed vertically downwards.

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

Acceleration free fall usually denoted by a letter other than acceleration. But the acceleration of free fall and acceleration are one and the same physical quantity and they have the same physical meaning. They participate equally in formulas for uniformly accelerated motion.

We write the “+” sign in 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. 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 bodies fall (gravity), caused by the Earth's gravitational field, is different for these bodies. For a stone it is larger (since the stone has more mass), for a feather it is smaller. But there is no dependence: 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 bedside table will move faster. And in order for the closet and the bedside table to move equally, the closet must be influenced more strongly than the bedside table. The Earth does the same. It attracts heavier bodies with greater force than lighter ones. And these forces are distributed between the masses in such a way that they all fall in a vacuum at the same time, regardless of mass.

Let us separately consider the issue of emerging air resistance. Let's take two identical sheets of paper. We will crumple one of them and at the same time let go of them. A crumpled leaf will fall to the ground sooner. Here, the different falling times are not related to body weight and gravity, but are due to air resistance.

Consider a body falling from a certain height h without initial speed. If coordinate axis To direct the op-amp upward, aligning the origin of coordinates with the surface of the Earth, we obtain the main characteristics of this movement.

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

IMPORTANT! Since the rise of a body to its maximum height and the subsequent fall to ground level are absolutely symmetrical movements (with the same acceleration, just one slower and the other accelerated), then the speed with which the body lands will be equal to the speed with which it tossed 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 ground level. Thus, the entire flight time will be double the time of rise or fall. The speed of a body at the same level when rising and falling will also be the same.

The main thing to remember

1) Direction of acceleration during free fall of the body;
2) Numerical value of the acceleration of free fall;
3) Formulas

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

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

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

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Free fall is interesting, but at the same time quite complex issue, 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 at equal speeds, if there is no medium 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 speed and acceleration the same?” That is, it turns out that sometimes the teacher mechanically presents some 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. More convincing images are needed than reasoning using the formula g = F heavy / 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 its previous form). Then similar conclusions are drawn for three, four, etc. But behind the formulas, students do not see the real explanation. The formula remains, as it were, on its own, but life experience prevents you from agreeing with the teacher’s story. And no matter how much the teacher speaks or convinces, there will be no lasting knowledge, logically justified, leaving a deep imprint in the memory. Therefore, as experience shows, in such a situation a different approach is needed, namely, an impact on the emotional level - to surprise and explain. In this case, you can do without the cumbersome experiment with a Newton tube. Quite enough simple experiments, proving the influence of air on the movement of a body in any environment and funny theoretical arguments, which, on the one hand, with their clarity can interest many, and on the other hand, will allow you to quickly and efficiently master 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 us consider the content of the presentation in more detail, since it is 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. Heading
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 surface of the earth at the same time. Gravity vectors are, accordingly, of different lengths.
5. Gravity is proportional to mass: Fweight = mg. In addition to this statement, the slide shows two circles. One is red and the other is blue, which matches the color of the letters for gravity and mass on this slide. To demonstrate the meaning of the directly and inversely proportional relationship, these circles, when you click the mouse, simultaneously begin to increase or decrease by the same number of times.
6. Gravity is proportional to mass. But this time it is shown mathematically. The animation allows you to substitute the same factors into both the numerator and the denominator of the formula for accelerating free fall. These numbers are reduced (which is also represented in the animation) and the formula becomes the same. That is, here we prove to students theoretically that during 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 globe unequal: 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, you should survey the students on the content of the poem)

We don’t count the air and fly to the ground,
The speed is increasing, it’s already clear to me.
Every second everything is the same:
Everyone add “ten”, the Earth will help us.
I increase my speed by meters per second.
As soon as I reach the ground, maybe I’ll calm down.
I'm glad that I'm on time, knowing the acceleration,
Experience free fall.
But probably better next time
I'll climb the mountains, maybe to the Caucasus:
There will be less "g" there. The only problem is
You step down and the numbers again, as always,
They will run at a gallop and cannot be stopped.
At least, in general, the air will slow down.
No. Let's go to the Moon or Mars instead.
Experiments there are many times safer.
Less attraction - I found out everything myself,
So, it will be more interesting to jump there.

1. 11. 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 experience in the movement of bodies in airless space. The Newton tube is connected by a hose to the Komovsky pump. After sufficient vacuum has been created in the tube, the bodies in it (pellet, cork and feather) fall almost simultaneously.
2. Animation: “Falling bodies in a Newton tube.” Bodies: fraction, coin, cork, feather.
3. Consideration of the resultant forces applied to a body when moving in the 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, 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 same mass. One of them was crumpled. Sheets are falling 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 a Newton tube, showing the role of air in resisting the movement of bodies.
   We take a textbook and a sheet of paper, 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 with identical speeds and accelerations, if the influence of air resistance is removed for the sheet, that is, the sheet is placed on a book. If the bodies are lifted 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, one more example can be offered. 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 also differ by a factor of 2, nothing surprises us. This is during normal driving 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 falling bodies. Then it will be clear that there are no fundamental differences. If a body of mass m is pulled by one horse, then for a body 2m 2 horses are needed so that the second body does not lag behind the first and moves 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 less, the more massive body would not keep up with the smaller body. It’s just like looking at horse racing on the previous slide. Thus, when 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. Nobody would change their speed, equal to zero. We are simply too accustomed to gravity and no longer notice its role. That is why the statement about the equality of the acceleration of gravity for bodies of very different masses seems so strange to us.

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

Let there be two electric charge- big and small; Let us assume that in a given electric field a large charge moves faster. Let's connect 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 compound charge from the side electric field, there will be more forces that each charge experienced separately. However, this information is still not enough to determine the acceleration of a 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 compound charge.

But why didn't Galileo encounter similar difficulties when he discussed the fall of heavy and light bodies? How does the movement of mass in a gravitational field differ from the movement of a charge in an electric field? It turns out that there is no fundamental difference here. To determine the motion 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 at a given force. To determine the motion of a body in a gravitational field, it is also necessary to take into account two quantities: 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. For Galileo, one value was enough because he considered the gravitational charge to be equal to mass.

Physicists usually 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 given force, is called “inertial mass”. So, for example, the mass discussed in the special theory of relativity is inertial mass.

Let us characterize 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 bread that the Earth attracts to itself with force V 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, larger one, say, with a force of 2? kg? Because there is more bread in the second loaf than in 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 weight depends on heavy mass. Heavy mass is a characteristic of a body that determines its weight, or, in other words, heavy mass determines the amount of 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 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 a body is attracted by the Earth; we measure the weight of the body in relation to the Earth. We could just as easily 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 the weight of a body depends on the mass relative to which it is measured. Let's imagine that the astronauts, going to Mars, took with them a loaf of bread, which weighs 1 on Earth kg. Having weighed it on the surface of Mars, they will find that the weight of the loaf was equal to 380 G. The heavy mass of the bread did not change during the flight, but the weight of the bread decreased almost threefold. 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 satisfy you in exactly the same way, regardless of where it is eaten - on Earth or on Mars. From this example it is clear 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 a body (relative to the Earth) is numerically equal to 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 lineup. Here you can’t stop the train right away—you have to slow down longer. Why does it take different times 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 locomotive driver care about the weight of the train? All that matters to him is how much resistance the train has to reducing speed. Why should we assume that the train, which the Earth attracts more strongly, resists the change in speed more stubbornly? True, everyday observations show that this is so, but it may turn out that this is a pure coincidence. There is no logical connection between the weight of the train and the resistance it provides to changes in speed.

So, we cannot explain by the weight of a body (and, consequently, heavy mass) the fact that under the influence of identical forces one body obediently changes its speed, while another requires considerable time for this. We must look for a reason elsewhere. The property of a body to resist a change in speed is called inertia. We noted earlier that in Latin “inertia” means laziness, lethargy. If a body is “lazy,” that is, changes its speed more slowly, then it is said to have greater inertia. We have seen that a train with less mass has less inertia than a train with more mass. Here we again used the word “mass”, but in a different sense. Above, mass characterized the attraction of a body by other bodies, but here it characterizes the inertia of a 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 heavy mass characterizes the gravitational influence on a body from other bodies, inertial mass characterizes the inertia of the body. If the heavy mass of a body doubles, then the force of attraction of it by other bodies will double. If the inertial mass doubles, then the acceleration acquired by the body under the influence of a given force will decrease by half. If, with an inertial mass twice as large, we demand 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 all bodies had inert mass equal to heavy mass? Let us have, for example, a piece of iron and a stone, and the inert mass of the piece of iron is three times greater than the inert mass of the stone. This means that in order to impart equal accelerations to these bodies, a piece of iron must be subject to a force three times greater than that applied to 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 greater than the heavy mass of a stone; a piece of iron will be attracted by the Earth three times stronger than a stone. But to impart equal accelerations exactly three times the force is required. Therefore, a piece of iron and a stone will fall to the Earth with equal accelerations.

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

Inert mass and heavy mass are different concepts that are logically unrelated. 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 have characterized the same property of the body using two different concepts. A body has only one mass. The fact that we previously attributed masses of two kinds to him was due simply to our insufficient knowledge of nature. It can now be rightfully said that heavy body mass is equivalent to inert mass. Consequently, the ratio of heavy and inert mass is to some extent similar 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 inertial and heavy mass. Since this equality has been established experimentally, one must certainly take into account the errors that inevitably appear in all measurements. According to Newton's estimate, for a body with heavy mass V 1 kg inert mass may differ from a kilogram by no more than 1 g.

The German astronomer Bessel used a pendulum to study the relationship between 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 claim that the inertial mass of a body is 1 kg may differ from the heavy mass by no more than 0.017 g. In 1894, the Hungarian physicist R. Eotvos managed to compare the inert and heavy masses with very great accuracy. From the measurements it followed that the inertial mass of the body V 1 kg may differ from the heavy mass by no more than 0.005 mG . Modern measurements made it possible to reduce the possible error by about a hundred times. Such accuracy of measurements makes it possible to assert that the inert 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 inert mass for the radioactive isotope of uranium. Uranium nuclei are unstable and over time turn into lead and helium nuclei. In this case, energy is released during the process of radioactive decay. 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 inert 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 is enough to simply say that the inert mass of a piece of uranium is 1 g. The same is its heavy mass. For radioactive bodies, the inert and heavy mass are also equal. The equality of inert and heavy mass is general property all bodies of nature.

For example, accelerators elementary particles, imparting energy to the particles, thereby increasing their weight. If, for example, electrons escaping from an 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, high-power electron accelerators are sometimes called electron “weighters.”)

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

So, if two bodies fall from the same height in a vacuum, then they will fall at the same time. 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, placed a pellet and a feather in it, and then pumped the air out of it. And it turned out that being in such a tube, both bodies fell down at the same time. The fact is that every body located 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 to a first approximation - yes. In fact, this is not uncommon in physics - read to the end).

If the fall occurs in the air, then in addition to the acceleration of free fall, one more thing arises; it is directed against the movement of the body (if the body is simply falling, 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 - F=ma, where a is acceleration). That is, if conventionally, bodies “fall” with the same acceleration, but “slow down” to varying degrees under the influence of the drag force of the medium. In other words, the foam ball will “slow down” more actively on the air as long as its mass is less than that of the lead ball flying nearby. In a vacuum there is no resistance and both balls will fall approximately (to the extent of the depth of vacuum and the accuracy of the experiment) simultaneously.

Well, in conclusion, the promised disclaimer. In the tube mentioned above, the same as Galileo’s, even under ideal conditions, the pellet will fall an insignificant number of nanoseconds earlier, again due to the fact that its mass differs insignificantly (compared to the mass of the Earth) from the mass of the feather. The point is that the law universal gravity, which describes the force of pairwise attraction between massive bodies, BOTH masses appear. That is, for each pair of such bodies, the resulting force (and therefore acceleration) will depend on the mass of the “falling” body. However, the contribution of the pellet to this force will be negligible, which means the difference between the acceleration values ​​for the pellet and the feather will be vanishingly small. If, for example, we talk about the “fall” of two balls of half and a quarter of the Earth’s mass, respectively, then the first one will “fall” noticeably earlier than the second. The truth is that it is difficult to talk about a “fall” 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 also the Earth at that moment approaches the stone at an insignificant (vanishingly) small distance. No comments. Just think about it before going to bed.