The concept of “movement” is not as easy to define as it may seem. From the everyday point of view, this state is the exact opposite of peace, but modern physics believes that this is not entirely true. In philosophy, movement refers to any changes that occur with matter. Aristotle believed that this phenomenon is equivalent to life itself. And for a mathematician, any movement of the body is expressed by the equation of motion recorded using variables and numbers.
Material point
In physics, the movement of various bodies in space is studied by a branch of mechanics called kinematics. If the dimensions of a certain object are too small in comparison with the distance that he has to overcome due to his movement, then he is considered here as a material point. An example of this is a car traveling along the road from one city to another, a bird flying in the sky, and much more. Such a simplified model is convenient when writing the equation of motion of a point for which a certain body is taken.
There are other situations. Imagine that the owner decided to move the same car from one end of the garage to the other. Here, the change in location is comparable to the size of the object. Therefore, each of the points of the car will have different coordinates, and it is considered as a three-dimensional body in space.
Basic concepts
It should be borne in mind that for a physicist, the path traveled by a certain object and movement are not at all the same thing, and these words are not synonyms. You can clarify the difference between these concepts by considering the movement of an airplane in the sky.
The trail that he leaves visually shows his trajectory, that is, a line. Moreover, the path represents its length and is expressed in certain units (for example, in meters). And movement is a vector connecting only the points of the beginning and end of the movement.
Similar can be seen in the figure below, which shows the route of a car traveling along a winding road and a helicopter flying in a straight line. The displacement vectors for these objects will be the same, and the paths and trajectories will be different.
Uniform movement 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 periods of time, the path that he walks over a given period does not change in magnitude.
What do we need to describe this movement of the body, or rather, the material point, as it was already agreed to call it? 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 = x 0 + v x t. It will describe the process in a general way.
An important concept when changing a body’s location is speed. In physics, it is a vector quantity, and therefore takes a positive and negative value. It all depends on the direction, because the body can move along the selected axis with increasing coordinate and in the opposite direction.
Movement relativity
Why is it so important to choose a coordinate system, as well as a reference point to describe the specified process? Just because the laws of the universe are such that without all this, the equation of motion will not make sense. This has been shown by such great scholars as Galileo, Newton and Einstein. From the beginning of life, being on Earth and intuitively accustomed to choosing it for the frame of reference, a person mistakenly believes that there is peace, although there is no such state for nature. A body can change its location or remain static only with respect to any object.
Moreover, the body can move and be at the same time at rest. An example of this is the train passenger’s suitcase, which lies on the top shelf of a compartment. He moves relative to the village, past which the train passes, and rests in the opinion of his master, who is located in the lower seat by the window. The cosmic body, once having received the initial speed, is capable of flying in space for millions of years, until it collides with another object. His movement will not stop because it moves only relative to other bodies, and in the frame of reference associated with it, the space traveler is at rest.
Equation Example
So, we will choose a point A for the reference point, and let the highway located nearby be the coordinate axis for us. And its direction will go from west to east. Suppose that a traveler set off on foot in point B, located 300 km away, at a speed of 4 km / h.
It turns out that the equation of motion is given in the form: x = 4t, where t is the travel time. According to this formula, it becomes possible to calculate the location of the pedestrian at any necessary time. It becomes clear that in an hour it will cover 4 km, in two - 8 and will reach point B after 75 hours, since its coordinate x = 300 will be at t = 75.
If the speed is negative
Suppose now that a car is traveling from B to A at a speed of 80 km / h. Here the equation of motion has the form: x = 300 - 80t. This is true, because x 0 = 300, and v = -80. It should be noted that the speed in this case is indicated with a minus sign, because the object moves in the negative direction of the 0X axis. How long does the car reach its destination? This will happen when the coordinate takes a zero value, that is, when x = 0.
It remains to solve the equation 0 = 300 - 80t. We get that t = 3.75. This means that the car will reach point B in 3 hours 45 minutes.
It must be remembered that the coordinate can also be negative. In our case, this would have turned out if there had been a certain point C, located in the western direction from A.
Movement with increasing speed
An object can move not only at a constant speed, but also change it over time. The movement of the body can occur according to very complex laws. But for simplicity, one should consider the case when the acceleration increases by a certain constant value, and the object moves in a straight line. In this case, they say that this is an equally accelerated movement. Formulas describing this process are given below.
Now consider specific tasks. Suppose that a girl sitting on a sled at the top of a mountain, which we choose for the beginning of an imaginary coordinate system with an axis along the downward slope, begins to move under the influence of gravity with an acceleration of 0.1 m / s 2 .
Then the equation of motion of the body has the form: s x = 0.05t 2 .
Understanding this, you can find out the distance a girl will ride on a sled for any of the moments of movement. In 10 seconds it will be 5 m, and in 20 seconds after the start of the movement downhill, the path will be 20 m.
How to express speed in the language of formulas? Since v 0 x = 0 (after all, the sled began to roll from a mountain without an initial speed only under the influence of gravity), then the record will not be too complicated.
The equation of speed of movement will take the form: v x = 0,1t. From it we can find out how this parameter changes over time.
For example, after ten seconds v x = 1 m / s 2 , and after 20 s it will take the value 2 m / s 2 .
If acceleration is negative
There is another type of displacement related to the same type. This movement is called equidistant. In this case, the speed of the body also changes, but over time does not increase, but decreases, and also by a constant value. Again we give a concrete example. The train, traveling before this at a constant speed of 20 m / s, began to slow down. Moreover, its acceleration was 0.4 m / s 2 . For the solution, we will take the starting point of the train path where it started to slow down, and we will direct the coordinate axis along the line of its movement.
Then it becomes clear that the motion is given by the equation: s x = 20t - 0.2t 2 .
And the speed is described by the expression: v x = 20 - 0.4t. It should be noted that a minus sign is placed before acceleration, since the train is braking, and this value is negative. From the obtained equations it is possible to conclude that the train will stop after 50 seconds, having traveled 500 m.
Difficult movement
To solve problems in physics, simplified mathematical models of real situations are usually created. But the multifaceted world and the phenomena occurring in it, do not always fit into such a framework. How to make an equation of motion in complex cases? The problem is solvable, because any complicated process can be described in stages. To clarify, we give an example again. Imagine that when the fireworks were launched, one of the rockets that took off from the ground at an initial speed of 30 m / s, reaching the top of its flight, exploded into two parts. The mass ratio of the resulting fragments was 2: 1. Further, both parts of the rocket continued to move separately from one another in such a way that the first flew vertically upward at a speed of 20 m / s, and the second immediately fell down. You should find out: what was the speed of the second part at the moment when it reached the ground?
The first step in this process will be the flight of a rocket vertically upward at an initial speed. Moving will be equally slow. In the description it is clear that the equation of motion of the body has the form: s x = 30t - 5t 2 . Here we assume that gravity acceleration is rounded up to a value of 10 m / s 2 for convenience. The speed will be described by the following expression: v = 30 - 10t. According to these data, it is already possible to calculate that the height of the lift will be 45 m.
The second stage of the movement (in this case, the second fragment) will be the free fall of this body at the initial speed obtained at the time of the disintegration of the rocket into pieces. In this case, the process will be uniformly accelerated. To find the final answer, he first computes v 0 from the law of conservation of momentum. The masses of bodies are 2: 1, and the speeds are inversely related. Consequently, the second fragment will fly down with v 0 = 10 m / s, and the velocity equation will take the form: v = 10 + 10t.
We learn the fall time from the equation of motion s x = 10t + 5t 2 . We substitute the already obtained value of the height of the lift. As a result, it turns out that the speed of the second fragment is approximately equal to 31.6 m / s 2 .
Thus, dividing a complex movement into simple components, it is possible to solve any complicated problems and make equations of motion of all kinds.