The equation of motion of the body. All kinds of the equations of motion


2018-03-18 15:41:22




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The Concept of “traffic” to determine not so simple as it may seem. From the worldly point of view, this state is the complete opposite of rest, but modern physics considers that it is not so. In philosophy the movement refers to any changes that occur with matter. Aristotle believed that this phenomenon is tantamount to life itself. But to a mathematician, any movement of the body is expressed by the equation of motion, written using variables and numbers.

Equation of motion

A point

In physics, the movement of different bodies in space studies the branch of mechanics called kinematics. If the dimensions of some object are too small compared to the distance that he has to overcome as a result of its movement, it is here considered as a material point. An example is a car traveling on the road from one city to another, a bird flying in the sky, and much more. Such a simplified model is convenient when writing equations of motion of a point, which is taken by some body.

There Are other situations. Imagine that the same car the owner decided to move from one end of the garage to another. Here the change of location is comparable with the size of the object. So each point will have different coordinates, and he is regarded as a solid body in space.

Basic concepts

Note that for physics the path traveled by a particular object, and moving – is not the same, and these words are not synonyms. To understand the difference between these concepts is to look at the movement of aircraft in the sky.


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Equation of motion of a body has the form

The Trail he leaves, shows its trajectory, that is the line. Here, the path is its length, and is expressed in certain units (e.g. in meters). And the movement – it is the vector that connects only the start and end points of the movement.

This can be seen in the figure below, which shows the route the traffic on the winding road, and a helicopter flying in a straight line. Vectors of displacement for these objects will be the same, but the path and trajectory –

Equation of motion

Uniform motion in a straight line

Now consider the various types of equations of motion. And let's start with the simplest case, when an object moves in a straight line at the same speed. This means that, after equal intervals of time the path that it traverses in a given period does not change the value.

We need to describe the motion of a body or rather of a material point, as has been agreed to call him? It is important to choose a coordinate system. For simplicity, we assume that the movement occurs along a certain axis 0X.

Then the equation of motion: x = x0 + vXT. It will describe the process in General.

An Important concept of changing the location of the body is speed. In physics it is a vector quantity, and so takes positive and negative value. It all depends on direction, because the body can move along the selected axis with increasing coordinate in the opposite direction.

Relative movement

Why is it important to choose a coordinate system and a reference point to describe this process? Just because the laws of the universe are such that without it the equation of motion will not make sense. It shows such great scientists as Galileo, Newton and Einstein. Since the beginning of life being on Earth and being accustomed to intuitively choose her for a reference system, one falsely believes that there is peace, though to nature there is no such condition. The body can change location or to remain static only with respect to any object.

Moreover, the body can move and be at rest simultaneously. An example of this can be a suitcase of a passenger train that is on the top shelf coupe. It moves relative to the village, past which passes through the composition, and rests according to its owner, is located on the lower seat by the window. Outer body, once having received the initial velocity, and is able to fly in space millions of years, until they encounter another object. The movement will not stop because it only moves relative to other bodies, and in the reference system associated with it, the space traveler is at rest.

equations of motion

Example of equation writing

So, choose a starting point a certain point A, in this case the coordinate axis let us motorway nearby. And the direction it will pass from West to East. Suppose that in this direction to point b, located over 300 km on foot traveler at a speed of 4 km/h.

It Turns out that the equation of motion is given as: x = 4t, where t-travel time. According to this formula, it is possible to calculate the location of the pedestrian at any moment. It becomes clear that in an hour it will be 4 km, two – 8 and reaches point B after 75 hours, as its coordinate x = 300 will be at t = 75.

If speed is negative

Suppose now that from in In And car rides, having a speed of 80 km/h. Here the equation of motion has the form: x = 300 – 80t. This is true, because x0 = 300 and v = -80. You should pay attention that the speed in this case is indicated with the sign "minus" because the object moves in the negative direction of axis 0X. After a time the vehicle reaches the destination? This happens when the coordinate will take the value zero, that is, when x = 0.

It Remains to solve the equation 0 = 300 – 80t. We find that t = 3,75. This means that the car will reach the point In 3 hours 45 minutes.

It Must be remembered that the coordinate can also be negative. In our case it would have been if there was a point, in a westerly direction from A.

Movement speed increase

To Move the object is not only constant speed, but change it over time. The movement of the body can occur by very complex laws. But for simplicity, consider the case when the acceleration increases at a certain constant value, and the object moves in a straight line. In this case, we say that is uniformly accelerated motion. The formula describing this process is given below.

Equation of motion x

Now consider a specific task. Let's say that the girl sat on the sled on top of a mountain, which we choose for the beginning of an imaginary coordinate system with the axis by the slope of the down, begins to move under gravity with an acceleration of 0.1 m/s2.

Then the equation of motion of a body has the form: sX = 0,05 t2.

Realizing this, you can see the distance that the girl will pass on the sled, for any of the displacement. After 10 seconds it will be 5 m, and 20 seconds after the beginning of the movement downhill path is 20 m.

How to Express speed on the language of formulas? Because v0X = 0 (because the sled began to go down without initial velocity only under the force of gravity), it is not too complicated.

The speed Equation of motion takes the form: vX= 0,1 t. From it we can learn how to change this parameter over time.

For example, after ten seconds, vX= 1 m/s2, and through 20 will be set to 2 m/s2.

the Equation of speed

If the acceleration is negative

There is another kind of displacement, belonging to the same type. This movement is called daunosamine. In this case, the velocity of the body is also changed, but with the passage of time does not increases and decreases, and also by a constant value. Again we give a concrete example. The train, going to a constant speed of 20 m/s, started to slow down. The acceleration was 0.4 m/s2. For the solution we take for the origin point of the train track where he started to slow down, and the coordinate axis will direct the line of travel.

Then it becomes clear that the movement is given by the equation: sX = 20t - 0.2 t2.

And the speed is described by the expression: vX = 20 – 0,4 t. It should be noted that before the acceleration is a minus sign as the train slows down, and the value is negative. From the obtained equations it is possible to conclude that the composition will stop after 50 seconds, traveling at 500 m.

Equation of motion has the form


For solving problems in physics are usually created simplified mathematical models of real situations. But the multi-faceted world and phenomena occurring in it do not always fit into such a framework. How to write the equation of motion in complex cases? The problem is solved, because any intricate process can be described in stages. To clarify again, here is an example. Imagine that when you run the fireworks one of the rockets taking off from the ground with an initial velocity of 30 m/s, reaching the top of his flight exploded in two parts. The ratio of the masses of the resulting fragments consist of...

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Alin Trodden - author of the article, editor
"Hi, I'm Alin Trodden. I write texts, read books, and look for impressions. And I'm not bad at telling you about it. I am always happy to participate in interesting projects."

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