Motion of the body under the action of gravity: definition, formulas

The motion of the body under the action of gravity is one of the central themes in dynamic physics. Even a regular schoolboy knows that the dynamics section is based on Newton's three laws . Let's try to disassemble this topic thoroughly, and an article detailing each example will help us make the study of body motion under the influence of gravity as useful as possible.

A bit of history

From time immemorial people watched with curiosity various phenomena occurring in our life. For a long time mankind could not understand the principles and structure of many systems, however, a long way of studying the world around us led our ancestors to a scientific revolution. Nowadays, when technologies are developing at an incredible speed, people almost do not think about how these or other mechanisms work.

Meanwhile our ancestors were always interested in the riddles of natural processes and the organization of the world, sought answers to the most difficult questions and did not stop studying until they found answers to them. So, for example, the famous scientist Galileo Galilei in the 16th century asked questions: "Why do bodies always fall down, what kind of force attracts them to the earth?" In 1589, he put a number of experiments, the results of which were very valuable. He studied in detail the patterns of free fall of various bodies, dropping objects from the famous tower in the city of Pisa. The laws that he derived were improved and described in more detail by the formulas of another famous English scientist, Sir Isaac Newton. He owns three laws on which almost all modern physics is based.

The fact that the laws governing the motion of bodies, described more than 500 years ago, are relevant to this day, means that our planet obeys unaltered laws. Modern man needs to at least superficially study the basic principles of the arrangement of the world.

Basic and auxiliary concepts of dynamics

In order to fully understand the principles of such a movement, you should first familiarize yourself with certain concepts. So, the most necessary theoretical terms:

• Interaction - is the impact of bodies on each other, in which there is a change or the beginning of their movement relative to each other. There are four types of interaction: electromagnetic, weak, strong and gravitational.
• Speed is a physical quantity that indicates the speed with which the body moves. Speed is a vector, that is, it has not only a value, but also a direction.
• Acceleration - the value that shows us the speed of change in the speed of the body in a period of time. It is also a vector quantity.
• The path of the path is a curve, and sometimes a straight line that the body delineates when moving. With uniform rectilinear motion, the trajectory can coincide with the displacement value.
• The path is the length of the trajectory, that is exactly as much as the body has passed for a certain amount of time.
• An inertial frame of reference is the environment in which Newton's first law holds, that is, the body retains its inertia, provided that all external forces are completely absent.

The above-mentioned concepts are quite enough to correctly draw or represent in the head the modeling of the body's motion under the influence of gravity.

What is power?

Let's move on to the basic concept of our topic. So, force is a quantity whose meaning lies in the impact or influence of one body on another quantitatively. And the force of gravity is the force that acts absolutely on every body that is on or near our planet. The question arises: where does this very power come from? The answer lies in the law of universal gravitation.

And what is the force of gravity?

On any body from the Earth is affected by gravitational force, which gives him some acceleration. Gravity always has a vertical direction down to the center of the planet. In other words, gravity attracts objects to the Earth, which is why objects always fall down. It turns out that gravity is a special case of the force of universal gravitation. Newton derived one of the main formulas for finding the force of attraction between two bodies. It looks like this: F = G * (m ₁ x M ₂ ) / R ² .

What is the acceleration of gravity?

The body, which was released from a certain height, always flies down under the force of attraction. The motion of the body under the action of gravity vertically up and down can be described by equations, where the basic constant is the acceleration value "g". This value is due solely to the action of the attractive force, and its value is approximately equal to 9.8 m / s ² . It turns out that the body, thrown from a height without the initial velocity, will move down with the acceleration equal to the value of "g".

Motion of the body under the action of gravity: formulas for solving problems

The basic formula for finding the gravitational force is as follows: F _gravity = m x g, where m is the mass of the body on which the force acts, and "g" is the acceleration of gravity (for simplicity, it is usually assumed to be 10 m / s ² ) .

There are several more formulas used to find one or another unknown when the body moves freely. So, for example, in order to calculate the path traveled by the body, it is necessary to substitute the known values in this formula: S = V ₀ x T + a x t 2/2 (the path is equal to the sum of the products of the initial velocity multiplied by the time and the acceleration by the square of the time divided by 2).

Equations for describing the vertical motion of the body

The motion of the body under the action of gravity along the vertical can be described by an equation that looks like this: x = x ₀ + v ₀ x t + a x t ² / 2. Using this expression, you can find the coordinates of the body at a known point in time. You just need to substitute the values known in the task: the initial location, the initial speed (if the body is not just released but pushed with some force) and acceleration, in our case it will be equal to the acceleration g.

In the same way, you can find the speed of the body, which moves under the force of attraction. The expression for finding an unknown quantity at any time: v = v ₀ + g x t (the initial velocity value can be zero, then the speed will be equal to the product of the acceleration of gravity by the time value for which the body makes movement).

Motion of bodies under the action of gravity: problems and methods of their solutions

When solving many problems related to gravity, we recommend using the following plan:

1. Define for yourself a convenient inertial frame of reference, it is usually customary to choose the Earth, because it meets many requirements for ISO.
2. Draw a small drawing or drawing, which depicts the main forces acting on the body. The motion of the body under the action of gravity implies a sketch or a diagram that indicates in which direction the body moves, if an acceleration equal to g acts on it.
3. Then choose the direction for the projection of forces and the accelerations obtained.
4. Record unknown quantities and determine their direction.
5. Finally, using the above formulas to solve problems, calculate all unknown quantities by substituting the data into equations for finding the acceleration or the traversed path.

A ready solution to an easy problem

When it comes to such a phenomenon as the motion of a body under the influence of gravity, determining how to solve the task more practical can be difficult. However, there are a few tricks, using which you can easily solve even the most difficult task. So, let's look at the living examples, how to solve this or that problem. Let's start with an easy to understand task.

Some body was released from a height of 20 m without the initial speed. Determine how much time it will reach the surface of the earth.

Solution: we know the path traveled by the body, it is known that the initial velocity was 0. Also, we can determine that only gravity acts on the body, it turns out that this body motion under the action of gravity, and therefore we should use this formula: S = V ₀ x T + a x t 2/2. Since in our case a = g, after some transformations we obtain the following equation: S = g × t ² / 2. Now it only remains to express the time through this formula, we get that t ² = 2S / g. We substitute the known quantities (here we assume that g = 10 m / s ² ) t ² = 2 x 20/10 = 4. Consequently, t = 2 s.

So, our answer: the body will fall to the ground in 2 seconds.

The trick that allows you to quickly solve the problem is as follows: you can see that the described movement of the body in the above problem occurs in one direction (vertically down). It is very similar to an evenly accelerated motion, since no force acts on the body, except for the force of gravity (we neglect the force of air resistance). Due to this, you can use the light formula to find the path at an evenly accelerated motion, bypassing the images of the drawings with the arrangement of forces acting on the body.

An example of a solution to a more complex problem

And now let's see how it is better to solve problems on the motion of a body under the influence of gravity, if the body moves not vertically, but has a more complex character of displacement.

For example, the following problem. Some object of mass m moves with an unknown acceleration down the inclined plane, the friction coefficient of which is equal to k. Determine the acceleration value that is present when a given body moves, if the slope angle α is known.

Solution: You should use the plan described above. First of all draw a picture of the inclined plane with the image of the body and all forces acting on it. It turns out that it has three components: gravity, friction and the reaction force of the support. The general equation of the resultant forces looks like this: F _friction + N + mg = ma.

The main feature of the problem is the condition of inclination at an angle α. When projecting forces on the ox axis and the oy axis, it is necessary to take into account this condition, then we get the following expression: mg x sin α-F _friction = ma (for the ax axis) and N - mg x cos α = F _friction (for the oy axis) .

F _{friction is} easy to calculate by the formula for finding the frictional force, it is equal to k x mg (coefficient of friction multiplied by the product of the mass of the body and the acceleration of gravity). After all the calculations, it remains only to substitute the values found in the formula, we obtain a simplified equation for calculating the acceleration with which the body moves along the inclined plane.