Momentum and Its Conservation

Momentum Linear momentum of a body is defined as the product of the mass of the body in motion times its velocity. Velocity is a vector so linear momentum is also a vector and points in the same direction as the velocity vector. The common units for momentum are kg-m/s and slug-ft/s.

Momentum Linear momentum is based upon Newton’s second law.

Momentum so

Momentum When a net force is applied to a body, it causes the linear momentum of the body to change with time. Newton used the term “quantity of motion” in explaining his second law. F= ma works well as long as the mass remains constant, but it does not always. A rocket burning fuel for example.

Law of Conservation of Momentum If the total external force acting on a system is equal to zero, then the final value of the total momentum of the system is equal to the initial value of the total momentum of the system. The Law of Conservation of Momentum is actually a restatement of Newton’s 3rd Law.

Law of Conservation of Momentum According to Newton’s Third Law, whenever one body exerts a force on the other, the other exerts a force that is equal in magnitude but opposite in direction. Since the net force acting on the objects is still zero, the change in momentum is also zero.

Impulse Looking at Newton’s second law, F=Dp/Dt. Rearranging this equation we get F Dt= Dp. F Dt is called impulse.(J) Any impulse acting on a body changes the momentum of the body. The impulse-momentum equation is:

Collisions In One Dimension A perfectly elastic collision is a collision in which no kinetic energy is lost. This is usually only possible at atomic levels. An inelastic collision is a collision in which some kinetic energy is lost. This is a very common type of collision. A perfectly inelastic collision occurs when two objects stick together and the loss of kinetic energy is maximum.

Collisions In One Dimension For a perfectly elastic collision:

Collisions In One Dimension For a perfectly elastic collision: VA is approach velocity, Vs is separation velocity.

Collisions In One Dimension For an inelastic collision: e is the coefficient of restitution. For a perfectly inelastic collision e=0. Vs=0, thus the objects stick together. The value of e must fall between 0 and 1.

Collisions In One Dimension If an object, like a ball, collides with the earth, we can determine the coefficient of restitution based upon the height to which the object bounces. For a perfectly inelastic collision the objects stick together upon colliding.

Collisions In One Dimension Solving for final velocity: