The amount of motion, which is often called the momentum of the body, is a key physical quantity in Newtonian mechanics. We give the impulse formula in this article, explain its meaning, and also consider other quantities and laws associated with it.
The momentum of the body and Newton's second law
At the beginning of the XVII century, Galileo Galilei, studying the laws of mechanical motion of bodies, introduced into physics an important quantity, which he called the momentum. Subsequently, Isaac Newton used the achievements of the Italian scientist in the formulation of his laws. Both scientists understood the quantity of motion as the product of the inertial mass of the body and the linear velocity of its movement in space, that is:
p¯ = m * v¯.
The quantity p¯ is a vector, which is directed in exactly the same way as the speed v¯. The physical meaning of the momentum is the manifestation of the inertial properties of the body and its kinetic energy at the same time.
If we take the time derivative of p¯, then we get the second Newtonian law:
dp¯ / dt = m * dv¯ / dt = m * a¯ = F¯.
This equality can be rewritten as follows:
dp¯ = F¯ * dt.
This expression suggests that the force F¯ acting during the time dt leads to a change in the momentum by dp¯. On the right side of the equality is a quantity called the momentum of force. For this reason, the expression for the momentum p¯ was called the momentum formula.
Impulse conservation
The importance of the studied quantity lies in the fact that it retains its value in closed systems. This means that if only internal forces act on the system’s bodies, then the momentum between them is distributed as a result of internal interactions, but the total momentum does not change over time, that is:
p¯ = const if F¯ = 0.
Typical interactions are collisions of bodies, such as balls. Collisions can be absolutely elastic, absolutely plastic or have an intermediate character. Nevertheless, regardless of the type of collision, the total momentum is always conserved, which cannot be said for kinetic energy. It is preserved only in the case of elastic collisions. For any other type of interaction, part of it is spent on heating bodies, that is, it goes into thermal energy.
When problems are solved, the momentum conservation formula should be used componentwise, that is, in the process of internal interactions of bodies, each component of the total momentum remains unchanged.
Momentum
If the quantity p¯ considered above characterizes the linear displacement of the body, then the angular momentum L¯ is a property of the rotation system. For a material point of mass m, rotating around the axis at a distance r with speed v, the angular momentum formula has the form:
L¯ = [r¯ * p¯] = [r¯ * m * v¯].
Square brackets indicate a vector product. The direction L¯ is perpendicular to the plane of rotation of the point and is determined by the rule of the right hand.
Like p¯, the quantity L¯ is conserved if there are no external influences on the system (the moment of force is zero). If the expression above is further transformed using the angular velocity ω¯ and the moment of inertia I, then the law of conservation of angular momentum can be written in the form:
L¯ = I * ω¯ = const.
In action, this law can be observed when the skater spins on ice, grouping his body in various ways.
Ball problem example
Using the momentum formula, knowing the law of its conservation, we will solve such a problem. Suppose that two balls of the same mass m fly at each other. The speed of the first ball is v 1 , the second - v 2 . It is necessary to determine the speed of the balls after the collision, knowing that v 1 = 2 * v 2 . The impact can be considered absolutely inelastic.
After the collision, two balls will move as a single body, therefore the following momentum formula is valid:
m * v 1 - m * v 2 = 2 * m * u.
For the positive direction of speed, the flight direction of the first ball is selected. Expressing u from here, and substituting the equality v 1 = 2 * v 2 , we obtain:
u = 1/2 * (v 1 - v 2 ) = v 2/2.
Thus, after the collision, two bodies will move in the direction of motion of the first body with a speed v 2/2.