The concept of "movement" is not as easy to define as it might seem. From an everyday point of view, this state is the complete opposite of rest, 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 tantamount to life itself. And for a mathematician, any movement of the body is expressed by an equation of motion written 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 an object are too small in comparison with the distance that it has to overcome due to its movement, then it is considered here as a material point. An example of this is a car driving 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 the equation of motion of a point, which is taken as a certain body.
There are other situations. Imagine that the owner of the same car decided to movefrom 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 will be considered as a three-dimensional body in space.
Basic concepts
It should be taken into account that for a physicist the path traveled by a certain object and movement are not the same thing at all, and these words are not synonyms. You can understand the difference between these concepts by considering the movement of an aircraft in the sky.
The trace it leaves clearly shows its trajectory, that is, the line. In this case, the path represents its length and is expressed in certain units (for example, in meters). And displacement is a vector connecting only the points of the beginning and end of the movement.
This can be seen in the figure below, which shows the route of a car traveling on a winding road and a helicopter flying in a straight line. The displacement vectors for these objects will be the same, but the paths and trajectories will be different.
Uniform movement in a straight line
Now consider different kinds of equations of motion. And let's start with the simplest case, when an object moves in a straight line with the same speed. This means that after equal periods of time, the path that he travels over a given period does not change in magnitude.
What do we need to describe this movement of a body, or rather, a material point, as it has already been agreed to call it? Important to choosecoordinate system. For simplicity, let's assume that the movement occurs along some axis 0X.
Then the equation of motion is: x=x0 + vxt. It will describe the process in general terms.
An important concept when changing the location of the body is speed. In physics, it is a vector quantity, so it takes on positive and negative values. Everything here depends on the direction, because the body can move along the selected axis with an 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 for describing the specified process? Simply because the laws of the universe are such that without all this, the equation of motion would not make sense. This is shown by such great scientists as Galileo, Newton and Einstein. From the beginning of life, being on the Earth and intuitively accustomed to choose it as a frame of reference, a person mistakenly believes that there is peace, although such a state does not exist for nature. The body can change location or remain static only relative to some object.
Moreover, the body can move and be at rest at the same time. An example of this is the suitcase of a train passenger, which lies on the top shelf of a compartment. He moves relative to the village, past which the train passes, and rests, according to his master, who is located on the lower seat by the window. The cosmic body, having once received the initial speed, is able to fly in space for millions of years, until it collides with another object. His movement will notstop 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, let's choose some point A as the starting point, and let the coordinate axis be the highway nearby. And its direction will be from west to east. Assume that a traveler sets out on foot in the same direction to 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 a pedestrian at any necessary moment. It becomes clear that in an hour he will travel 4 km, in two - 8 and will reach point B after 75 hours, since his 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 x0 =300, and v=-80. Please note that the speed in this case is indicated with a minus sign, because the object is moving in the negative direction of the 0X axis. How long will it take for the car to reach its destination? This will happen when the coordinate becomes zero, 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 and 45 minutes.
It must be remembered that the coordinate can also be negative. In our case, this would be if there were some point C, located in the western direction from A.
Moving 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, we should consider the case when the acceleration increases by a certain constant value, and the object moves in a straight line. In this case, we say that this is uniformly accelerated motion. The formulas describing this process are given below.
And now let's look at specific tasks. Suppose that a girl, sitting on a sled on top of a mountain, which we will choose as the origin of an imaginary coordinate system with the axis directed downward, begins to move under the influence of gravity with an acceleration equal to 0.1 m/s2.
Then the equation of motion of the body is: sx =0, 05t2.
Understanding this, you can find out the distance that the girl will travel on the sled for any of the moments of the movement. After 10 seconds it will be 5 m, and 20 seconds after the start of the downhill movement, the path will be 20 m.
How to express speed in formula language? Because v0x =0), then the recording will not be too difficult.
The motion velocity equation will take the form: vx=0, 1t. From it wewill be able to see how this parameter changes over time.
For example, after ten seconds vx=1 m/s2, and after 20 s it will take the value 2 m/s 2.
If acceleration is negative
There is another kind of movement that belongs to the same type. This movement is called equally slow. In this case, the speed of the body also changes, but over time it does not increase, but decreases, and also by a constant value. Let's take a concrete example again. The train, which had previously been traveling at a constant speed of 20 m/s, began to slow down. At the same time, its acceleration was 0.4 m/s2. For the solution, let's take as the origin the point of the train's path, where it started to slow down, and direct the coordinate axis along the line of its movement.
Then it becomes clear that the movement is given by the equation: sx =20t - 0, 2t2.
And the speed is described by the expression: vx =20 – 0, 4t. It should be noted that a minus sign is placed before the acceleration, since the train slows down, and this value is negative. From the equations obtained, it is possible to conclude that the train will stop after 50 seconds, having traveled 500 m.
Complex movement
To solve problems in physics, simplified mathematical models of real situations are usually created. But the multifaceted world and the phenomena taking place in it do not always fit into such a framework. How to write an equation of motion in complexcases? The problem is solvable, because any confusing process can be described in stages. To clarify, let's take an example again. Imagine that when launching fireworks, one of the rockets that took off from the ground with an initial speed of 30 m/s, having reached the top point of its flight, broke into two parts. In this case, 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 upwards at a speed of 20 m / s, and the second immediately fell down. You should know: what was the speed of the second part at the moment it hit the ground?
The first stage of this process will be the flight of the rocket vertically upwards with the initial speed. The movement will be equally slow. When describing, it is clear that the equation of motion of the body has the form: sx=30t – 5t2. Here we assume that the gravitational acceleration is rounded up to 10 m/s for convenience2. In this case, the speed will be described by the following expression: v=30 – 10t. Based on 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 already the second fragment) will be the free fall of this body with the initial speed obtained at the moment the rocket breaks apart. In this case, the process will be uniformly accelerated. To find the final answer, first calculates v0 from the law of conservation of momentum. The masses of bodies are in a ratio of 2:1, and the velocities are inversely related. Therefore, the second fragment will fly down from v0=10 m/s, and the velocity equation becomes: v=10 + 10t.
We learn the fall time from the equation of motion sx =10t + 5t2. Substitute the already obtained value of the lift height. As a result, it turns out that the speed of the second fragment is approximately 31.6 m/s2.
Thus, by dividing a complex movement into simple components, you can solve any intricate problems and make equations of motion of all kinds.
