1. You are well aware that bodies fall to the ground if they are not held by a support, a suspension thread, a hand, etc. When a body falls, its speed increases, i.e., the fall of bodies is an accelerated movement.
If we simultaneously release metal and paper circles of the same size from our hands from a certain height and observe their movement, then we will notice that the metal circle will fall to the ground before the paper one. It can be assumed that the time of falling bodies depends on their mass. To verify this, let's take two identical sheets of paper, crumple one of them and simultaneously release them from our hands. The crumpled piece of paper will fall to the ground first. Therefore, different fall times are not related to the mass of the bodies.
Obviously, a crumpled sheet of paper and a smooth one experience different air resistance when falling. This assumption can be confirmed experimentally.
Take a thick-walled tube, one end of which is sealed, and the other is equipped with a tap. A pellet, a piece of cork, and a bird's feather are placed in the tube (Fig. 33). If you quickly turn the tube over, then these bodies will fall to its bottom. You can see that the pellet will fall before everyone else, and the feather - after all the bodies. If now the air is pumped out of the tube and, having closed the tap, it is turned over again, then all three bodies will reach the bottom of the tube at the same time, despite the fact that they have different shapes and masses. Therefore, all bodies in airless space (in vacuum) fall with the same acceleration, which is called free fall acceleration.
The fall of bodies in airless space is called free fall.
2. The free fall of bodies is a uniformly accelerated motion.
The free fall acceleration is always directed towards the center of the Earth and has the same value for all bodies at their same initial position relative to the Earth's surface.
Indeed, as you already know, the modulus of displacement of a body during uniformly accelerated motion without an initial velocity is calculated by the formula: s= . From the experiment described above, it follows that a pellet, a piece of cork, and a bird's feather make the same movements in the same intervals of time, so they all move with the same acceleration.
A body thrown vertically upwards also moves uniformly with the acceleration of free fall. In this case, the velocity and acceleration vectors of the body are directed in opposite directions, and the velocity modulus decreases with time.
3. Free fall acceleration is denoted by the letter g. As you know from the 7th grade physics course, the acceleration of free fall depends on the geographical latitude of the area. At the latitude of Moscow near the Earth's surface, it is equal to 9.81 m/s 2 . When solving problems, if high accuracy of the result is not required, take g\u003d 10 m / s 2.
The free fall acceleration depends on the height of the body above the Earth's surface. The higher the body is raised, the weaker it is attracted to the Earth and the lower the acceleration of free fall. For example, for passenger aircraft with a maximum altitude of about 10 km above sea level, the acceleration due to gravity at this altitude is 9.78 m/s 2 . For the heights at which modern fighters fly, a more significant decrease in the acceleration of free fall is characteristic. So, at an altitude of 18 km, it is equal to 9.72 m / s 2.
The acceleration of gravity is even less important at altitudes where the orbits of artificial Earth satellites and space stations are located. Thus, the maximum height of the first artificial Earth satellite relative to sea level was 947 km. The free fall acceleration at this height is 7.41 m/s 2 .
4 * . Free fall was studied by an Italian scientist, one of the founders of classical mechanics, Galileo Galilei (1564-1642) at the end of the 16th century. He dropped from the Leaning Tower of Pisa at the same time a ball weighing about 200 g and a body weighing 40 kg, which has a cigar shape. Contrary to the opinion that existed at that time, the bodies reached the surface of the Earth almost simultaneously. The ball was only a few centimeters behind. Galileo did not have precise instruments for measuring time, he used hourglass, so the value of the acceleration of free fall was measured by him with a large error. In particular, in his work “Dialogue on the two main systems of the world - Ptolemaic and Copernican”, Galileo argued that bodies fell from a height of 60 m for 5 s and, based on these data, obtained the value of the free fall acceleration almost 2 times less than that obtained in present time.
To improve the accuracy of the experiment on the study of uniformly accelerated motion and free fall, in particular, Galileo studied the sliding of balls from an inclined plane. He experimentally established the proportionality of the path traversed by the ball to the square of time and the law of the ratio of the paths traversed by it in successive equal intervals of time.
5. Problem solution example
Two bodies simultaneously begin to move: one vertically upwards with a speed of 20 m/s, the other vertically downwards from a height of 60 m without initial velocity. Determine the time and coordinate of the meeting point of the bodies.
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Given: |
Solution |
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v 01 = 20 m/s v 02 = 0 h= 60 m g = 10 m/s 2 |
Let's connect the reference system with the Earth. For the origin of the coordinates, we take the point from which the first body was thrown from the surface of the Earth, the axis OYlet's direct it upwards, we will take the moment of throwing the bodies as the beginning of the countdown (Fig. 34). |
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t? y? |
We write the equation of motion in projections onto the axis OY:
y = y 0 + v 0y t + .
For the first body, this equation has the form:
y 1 = y 01 + v 01y t + .
Given that y 01 = 0; v 01y = v 01 ; g y = –g, we get
y 1 = v 01 t – .
Equation of motion of the second body:
y 2 = y 02 + v 02y t + .
Because the y 02 = h; v 02y = 0; g y = –g, then
y 2 = h – .
At the moment of the meeting of the bodies, their coordinates will be the same: y 1 = y 2 = y. Then v 01 t –= h – ; v 01 t = h.
Hence the time of the meeting of the bodies t = ;
t== 3 s.
We find the coordinate of the meeting place of the bodies from the equation of motion of the first body.
y= 20 m/s 3 s –= 15 m.
Answer: t= 3 s; y= 15 m.
Questions for self-examination
1. What movement is called free fall?
2. To what kind mechanical movement refers to free fall?
3. How to experimentally prove that the acceleration of free fall is the same for all bodies at a given point in space?
4. What does free fall acceleration depend on?
Task 8
1. A ball falls to the ground from a height of 20 m with an initial velocity of zero. How long will it take to reach the earth's surface? What is the speed of the ball when it hits the ground? At what height relative to the ground will the ball be in 1 second after the start of the fall? What speed will it have at this point in time? Ignore air resistance.
2. According to task 1, plot the dependency graphs of the velocity projection on the axis Y and the modulus of the ball's speed versus time, if the axis Y directed: a) vertically down; b) vertically up.
3. At what height relative to the Earth's surface will two balls meet if one is thrown vertically upward with a speed of 10 m/s, and the other falls from a height of 10 m without initial velocity? The balls start moving at the same time. What speed relative to the ground will the balls have at this height? Ignore air resistance. Build graphs of the dependence of the coordinates of each ball on time and determine from the graph the time and coordinate of the place of their meeting * .
4 * . Calculate the free fall acceleration using the data obtained by Galileo.
5. Build graphs of the dependence of the projection of the speed of bodies on time according to the problem considered in § 8 * . Based on this data, plot the dependence of the coordinates of each body on time and graphically determine the time and coordinate of the meeting point of the bodies.
Such an experience gives grounds to consider the movement of bodies along a curvilinear trajectory, having received speed at an angle to the horizon, as two independent movements - vertically and horizontally. Moreover, these movements proceed independently of each other and do not influence each other.
This statement is called the principle of independence of movement, extends to the motion of bodies thrown at an angle to the horizon.
Since the complex curvilinear motion of a falling body can be represented as the sum of two independent simple motions vertically and horizontally, for further reasoning we will dwell on the analysis of the body's motion only in the vertical direction. In this case, for simplicity, we will assume for the time being that the initial velocity of the body is zero.
Even the simplest observations give us reason to be convinced that the medium in which the falling body moves has a significant influence on the nature of the motion. First of all, air acts as such a medium.
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Indeed, let's drop a steel ball and a piece of paper from the same height. A piece of paper reaches the Earth's surface in much longer time than a ball. It may seem that this is due to the fact that the ball is more massive than a piece of paper. However, a crumpled piece of paper reaches the Earth's surface almost simultaneously with a steel ball. Probably, the results of the experiments can be explained by the resistance that air provides to falling bodies.
A piece of paper falling from the same height and a metal sheet equal in area to it, again, make the same movement in obviously different times. But, on the other hand, it is worth putting a paper sheet on top of a metal one, as it stops lagging behind the metal sheet during the fall.
After carrying out such experiments, it becomes almost obvious that the influence of air on falling bodies is significant.
It can be assumed that in an airless space, different bodies, regardless of their size, shape, substance from which they are made, under the same initial conditions, will fall in the same way.
This assumption can be verified by direct experiment. To do this, you can take a long, closed tube, in which, for example, a feather, a piece of paper, a pellet are placed. If the air is evacuated from the tube and the given objects are allowed to fall from the same height, one can be convinced of the validity of the put forward assumption.
A more precise experiment is also possible. For example, it is possible to directly measure the time of falling from the same height of several balls that differ significantly from each other in size and mass.
Within the measurement accuracy, this time turns out to be the same.
In its pure form, free fall is unlikely to be studied by us. But if we take into account that air has a relatively small effect on falling small metal balls, we will take their movement in air as a model of free fall.
Let us ask ourselves the question: does the speed of the body remain constant or does it change during the fall?
It is plausible to assume that the speed of a falling body increases in the process of motion.
Simple direct observations are unlikely to allow us to prove the validity of this hypothesis. However, circumstantial evidence suggests that this is the case. Such data include, for example, the sound of an impact, the height of the rebound of a metal ball falling on a wooden table from different heights.
If the speed of the falling body increases with time, then the question arises: is the acceleration of the falling body constant or not?
It is possible that free fall is a kind of uniformly accelerated motion. But it is also possible that the acceleration either increases or decreases as the motion progresses.
If we accept the first version as a working one, then we should measure the time of the fall of a body from different heights and in each case calculate from well-known formula expected acceleration. If the calculations made taking into account the measurement accuracy will give the same result, the version will find its experimental confirmation. Otherwise, alternative versions will need to be checked.
This experiment has been carried out many times. It turned out that the acceleration of gravity in a given area of the Earth, provided that the height above its surface (compared to the radius of the Earth) changes slightly, is a constant value. On average, the free fall acceleration near the Earth's surface is
An analysis of a stroboscopic photograph of the movement of a body thrown at an angle to the horizon shows that the movements made by the body in the horizontal direction for equal periods of time are equal to each other. This means that the body moves uniformly in this direction. Movements in the vertical direction, performed in the same equal time intervals, are not equal to each other.
On the ascending part of the trajectory, the displacements decrease, on the downward part they increase. This is due to the accelerated nature of the movement of the body. The symmetry of the curve indicates that the acceleration modulus remains constant throughout the entire trajectory.
Since the horizontal coordinate of a body thrown at an angle to the horizon changes over time according to a linear law, and along the vertical - according to a quadratic law, the trajectory of such a movement is a parabola.
Free fall of bodies is the fall of bodies to the Earth in a vacuum in the absence of interference. under the influence of gravity in the absence of air resistance can be considered free fall. For example, an athlete is in free fall, jumping into the water from a tower or a ball released from his hand.
In 1583 an Italian scientist Galileo Galilei(1564-1642) found that in the absence of air resistance, all bodies, regardless of their mass, fall to the ground with the same g, which is directed vertically downwards. This acceleration is called acceleration of gravity. With a free fall of a body from a small height h from the surface of the Earth (moreover, h is much less than the radius of the Earth R З, where Earth radius R З ~ 6000 km ) the force of attraction remains almost constant, so the acceleration of free fall also remains constant.
This conclusion is confirmed by the experiment with falling bodies in a glass tube from which air is pumped out (Fig. 1.24). A piece of lead, a light feather and a pellet reach the bottom of the tube at the same time. Therefore, they fall with the same acceleration.
Free fall can be considered as a special case of uniformly accelerated motion. Acceleration of free fall depends on the height above sea level and on the geographical latitude of the place. It varies from approximately 9.83 m/s 2 at the pole to 9.78 m/s 2 at the equator. At the latitude of Moscow, the free fall acceleration is assumed to be g = 9.8 m/s 2 . Therefore, in most cases, when solving problems in physics, the free fall acceleration is assumed to be 9.8 m/s 2 .
The difference in the value of acceleration is explained by the daily rotation of the Earth and the shape of the Earth - the Earth is flattened at the poles, so the pole radius of the Earth is less than the equatorial radius.
Rice. 1.24. Free fall of bodies. The dependence of the acceleration of free fall on the height above sea level can be obtained by applying Newton's second law and the law gravity. Free fall acceleration modulus is:
g = G(M /(R + h) 2)
where G is the gravitational constant (or the constant of universal gravitation), G = (6.673 ± 0.003) * 10 -11 n * m 2 / kg 2 M is the mass of the Earth, M = 5.9736 * 10 24 kg R is the radius of the Earth, average the radius of the Earth R З.СР = 6371 km, h is the height of the body above sea level (above the Earth's surface).
From this equation it can be seen that when the body is lifted, the acceleration of free fall decreases. This becomes noticeable when climbing to a height of more than 300 km.
In some areas the globe gravitational acceleration may differ from the value of acceleration at a given latitude. Such deviations are observed in places where there are deposits of minerals.
The movement of bodies along the vertical (up or down) near the surface of the Earth, without taking into account air resistance, is a rectilinear uniformly accelerated movement. When describing such a movement, the coordinate axis OY is chosen, directed up or down. Regardless of the direction of the OY axis, the gravitational acceleration vector is directed vertically downwards.
Formulas for calculating coordinates (or heights) and velocities will take the following form.
The speed of the body at any given time
v y = ± v oy ± g y t
body movement
s y = ± v oy t ± (g y t 2) / 2
Body coordinates (body height)
y = h = h 0 ± v oy t ± (g y t 2) / 2
The speed of the body at any point along the way
v y 2 \u003d v oy 2 + 2g y (h - h 0)
If the OY axis is directed downward, then the projection of the gravitational acceleration g y on this axis is positive. If the OY axis is directed upwards, then the projection g y is negative. For example, a ball thrown vertically up to the top point of the ascent moves uniformly slow, and its downward movement will be uniformly accelerated.
The projections of the initial vo oy and final v y velocities are positive if the direction of the velocities coincides with the direction of the OY axis, and negative if the directions of the OY axis and the velocities are opposite.
Tasks in mechanics (dynamics), on the topic
Movement under the influence of gravity in a vertical direction
From the manual: GDZ k problem book Rymkevich for grades 10-11 in Physics, 10th edition, 2006
Find the acceleration of free fall of the ball according to figure 31, made from a stroboscopic photograph. The interval between shots is 0.1 s, and the side of each square of the grid in the life-size photo is 5 cm
SOLUTION
In free fall, the first body was in flight 2 times longer than the second. Compare the final velocities of bodies and their displacements
SOLUTION
G. Galileo, studying the laws of free fall (1589), threw various objects without initial speed from an inclined tower in the city of Pisa, the height of which is 57.5 m. How long did objects fall from this tower and what was their speed when they hit the ground
SOLUTION
A swimmer, having jumped from a five-meter tower, plunged into the water to a depth of 2 m. How long and with what acceleration did he move in the water
SOLUTION
A body falls freely from a height of 80 m. What is its displacement in the last second of the fall
SOLUTION
How long did the body fall if it traveled 60 m in the last 2 s?
SOLUTION
What is the displacement of a freely falling body in the nth second after the start of the fall
SOLUTION
What initial speed must be given to a stone when it is thrown vertically down from a bridge 20 m high so that it reaches the surface of the water in 1 s? How long would a stone fall from the same height without initial velocity
SOLUTION
One body falls freely from a height h1; simultaneously with it, another body starts moving from a higher height h2. What should be the initial speed u0 of the second body so that both bodies fall at the same time
SOLUTION
An arrow fired from a bow vertically upwards fell to the ground after 6 s. What is the initial speed of the boom and the maximum lifting height
SOLUTION
How many times greater is the height of the body thrown vertically upwards on the Moon than on Earth, with the same initial velocity?
SOLUTION
By how many times must the initial velocity of a body thrown vertically upwards be increased so that the height of the lift is increased by 4 times
SOLUTION
From a point located at a sufficiently high altitude, two bodies are simultaneously thrown with the same modulus of velocities v0 = 2 m/s: one vertically upwards, and the other vertically downwards. What will be the distance between the bodies after 1 s; 5 s; after a period of time equal to
SOLUTION
When throwing the ball vertically upwards, the boy tells him the speed is 1.5 times greater than the girl. How many times higher will the ball thrown by the boy rise
SOLUTION
An anti-aircraft gun projectile fired vertically upwards at a speed of 800 m/s reached its target in 6 s. At what height was the enemy aircraft and what was the speed of the projectile when it reached the target? How different real values the desired values from the calculated
SOLUTION
A body is thrown vertically upward with a speed of 30 m/s. At what height and after what time the speed of the body (modulo) will be 3 times less than at the beginning of the ascent
SOLUTION
The BALL was thrown vertically upwards twice. The second time he was told the speed, 3 times greater than the first time. How many times higher does the ball rise on the second throw?
SOLUTION
A body is thrown vertically upward with a speed of 20 m/s. Write the equation of motion y = y(t). Find the time interval after which the body will be at a height of: a) 15 m; b) 20 m; c) 25 m. Indication. The Y axis is directed vertically upwards; accept that at t = 0 y = 0
SOLUTION
A ball is thrown vertically upwards from a balcony 25 m above the ground with a speed of 20 m/s. Write a formula for the dependence of the coordinate on time y(t), choosing as the origin: a) the point of throw; b) the surface of the earth. Find the time it takes for the ball to hit the ground.