One of the important physical characteristics that are used in the kinematic and dynamic description of the mechanical motion of bodies is momentum. In this article, we give a definition of the law of conservation of momentum in physics, and also demonstrate by the example of the problem how it can be used in practice.
Impulse or momentum
Before considering the formula and definition of the law of conservation of momentum, we will get acquainted with the quantity itself. In physics, a mechanical impulse is usually understood as the product of the inertial mass of a body and the linear velocity of its movement in space. Mathematically, the value is written as follows:
p¯ = m * v¯.
As can be seen, the momentum p¯ is a vector characteristic. Its direction coincides with the speed, and the module is m times larger than the module | v¯ |. The formula shows that p¯ contains information simultaneously on the kinetic and inertial properties of the body.
For the first time, scientists of the New Age began talking about the impulse. The first of these was Galileo Galilei. In one of his scientific works, he used this quantity to describe the motion of bodies. He called her the amount of movement. Subsequently, Isaac Newton took it into service when formulating the basic laws of mechanics.
Change in momentum
Recall that the first Newtonian law says that the body does not change its velocity in absolute value and direction until some external force begins to act on it. Since the mass of a moving body in mechanics also remains constant, the momentum p will not change.
As soon as some external force acts on the body, in accordance with the second Newtonian law, it will lead to the appearance of linear acceleration. Mathematically, we have the following equalities:
F = m * a = m * dv / dt = dp / dt =>
dp = F * dt.
We came to an interesting conclusion: it turns out that the magnitude of the change in the momentum will be equal to the product of force and its duration. This work is called the impulse of power. From here we get the name of the impulse for changing the momentum dp.
Determination of the law of conservation of momentum
So, we examined the physical quantity itself, showed under what conditions the momentum of one body can change. And what will happen to the system of moving bodies, if left to their own devices? Since the system is closed, that is, it is not affected by external forces, the total amount of motion of all bodies will not change in time, despite the fact that interactions between the bodies themselves can occur. The latter are elastic or plastic collisions of bodies. If during the movement of bodies in the system there are scattering forces, for example, a friction force, then the full impulse will not be preserved.
In accordance with the definition of the law of conservation of momentum, the mathematical formula for it is written in the form:
p¯ = const.
Since each component of p¯ remains unchanged, it is more convenient for practice to use the following system of equalities:
p x = const;
p y = const;
p z = const.
Note that the momentum component p i , where i = x, y, z, here means the sum of all the corresponding momentum components for each body that is part of the system under study.
In physics, problems with a one-dimensional or two-dimensional character of motion are usually encountered, therefore, to solve them, it suffices to consider a system of one or two equations.
Elastic and plastic collisions
Above, when considering the definition of the law of conservation of momentum, it was said that it is satisfied in elastic and plastic collisions. Let us explain this moment.
Imagine the movement of two balls towards each other. If, as a result of their collision, the total kinetic energy has not changed, then there was an elastic collision. Any decrease in kinetic energy indicates the presence of plastic deformation in bodies after a collision. A collision is called absolutely inelastic, after which two balls (bodies) begin to move as a whole.
To determine the degree of plasticity of the collision of two bodies, a special value is introduced, which is called the recovery coefficient. This coefficient is equal to the ratio of the difference in the velocities of the bodies after the collision to the difference in their velocities before the collision, which must be taken with the opposite sign. The coefficient varies from 0 to 1, and these numbers correspond to absolutely plastic and elastic collisions, respectively. The law of conservation of momentum remains valid for any value of the recovery coefficient.
Problem solving example
We will solve the following simple physical problem for applying the acquired knowledge. Suppose that two balls are moving towards each other. The masses of the balls are 5 kg and 3 kg. Their speeds are 4 m / s and 6 m / s. After a completely plastic collision, both balls begin to move together. It is necessary to determine the direction and speed of their joint movement.
To solve the problem, we make the following notation:
v 1 = 4 m / s, m 1 = 5 kg;
v 2 = 6 m / s, m 2 = 3 kg.
Suppose that the first ball moves from left to right, and the second in the opposite direction. Then we get:
m 1 * v 1 - m 2 * v 2 = (m 1 + m 2 ) * u =>
u = (m 1 * v 1 - m 2 * v 2 ) / (m 1 + m 2 ) = (5 * 4 - 3 * 6) / (5 + 3) = 2/8 = 0.25 m / s .
Since a positive velocity value is obtained, this means that both balls after the collision will move from left to right.
The law of conservation of angular momentum
The mathematical definition of the angular momentum of a material point can be given as follows:
L = p * r.
Where the letter r denotes the radius of rotation of the material point around the axis.
Like a linear impulse, the quantity L is conserved only when external forces are not acting on the system. Under the moment of force is assumed a value equal to the product of the force and the radius of its application relative to the axis of rotation. The corresponding conservation law is written as follows:
I * ω = const.
Where I is the moment of inertia, ω is the angular velocity.
People who like to watch figure skating have probably noticed this law in action more than once, when an athlete, changing his body position, begins to spin faster or slower on ice.