# The motion of the body under the action of gravity: the definition, formulas

The motion of the body under the action of gravity isone 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 to 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 curiosityvarious phenomena that occur in our lives. For a long time mankind could not understand the principles and structure of many systems, but 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.

And meanwhile our ancestors were always interestedpuzzles 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 series 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 of 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.friend, under 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 - this is a physical quantity, indicating 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 time interval. 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 passed for a certain amount of time.
- An inertial frame of reference is the medium in which Newton's first law holds, that is, the body retains its inertia, provided that all external forces are completely absent.

The above concepts are sufficient to correctly draw or present in the head the modeling of the body's motion under the action 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 gravity?

Any body on the Earth's side is affectedgravitational force, which gives him some acceleration. Gravity always has a vertical direction down to the center of the planet. In other words, the force of gravity attracts objects to the Earth, which is why objects always fall down. It turns out that the force of 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 an acceleration equal to the value of "g".

## Movement of the body under the action of gravity: formulas for solving problems

The basic formula for finding the gravity is as follows: F_{gravity }= m × g, where m is the mass of the body acting on the force, 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 forfinding one or another unknown with free movement of the body. 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 movement 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 certain 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 action of the force of attraction. The expression for finding an unknown quantity at any time: v = v_{0 }+ g x t (the value of the initial velocity can be equal to zero, then the speed will be equal to the product of the acceleration of gravity by the time value for which the body makes the motion).

## 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:

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

## A ready solution to an easy problem

When it comes to a phenomenon such as movementbodies under the action of gravity, the determination of how it is more practical to solve the task posed can be difficult. However, there are several 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.

Decision: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 (we assume that g = 10 m / s^{2}) t^{2} = 2 x 20/10 = 4. Therefore, t = 2 s.

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

A trick that allows you to quickly solve a problem isas 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 equally 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 the problem of the motion of the 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 value of the acceleration that exists when a given body moves, if the angle of inclination α is known.

Decision: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 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 x axis) and N - mg x cos α = F_{friction }(for the oy axis).

F_{friction} It is easy to calculate from the formula of finding the forcefriction, it is equal to k x mg (the 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 found values into the formula, we obtain a simplified equation for calculating the acceleration with which the body moves along the inclined plane.