Momentum and ImpulseAn object undergoing translational motion possesses linear momentum, p, a vector quan-tity, computed by taking the product of the mass and the velocity of the object.

The direction of the momentum vector is the same as thatof the velocity.

The Conservation of MomentumMomentum is a useful concept. In any mechanical system composed of mutually interact-ing objects, though not subject to external forces, the net linear momentum remainsunchanged. This is the law of the conservation of momentum. The law has two mathe-matical forms describing the two types of collisions which objects within such a system canhave with each other, elastic or inelastic collisions.

In an elastic collision, the total translational kinetic energy, not only the momentum, ofthe colliding objects remains unchanged before and after contact; none of the kinetic ener-gy is transformed into other types of energy, such as heat or vibrational energy.

The mathematical expression to describe conservation of momentum is as follows:

Because also the kinetic energy is conserved in an elastic collision, the following expres-sion also describes an elastic collision:

p

beforecollision

aftercollision

p1i p2i

p1f p2f+

+

=

=p1i p2i

p1f p2f

pi tot

pf tot

=

total momentum remains the samebefore and after collision.

© J.S Wetzel, 1993

An inelastic collision is characterized by a decrease in the translational kinetic energy ofthe objects upon collision. Some kinetic energy is transformed by the collision into otherforms of energy such as heat or the energy of sound waves. As with an elastic collision,

momentum is conserved in inelastic collisions, though not kinetic energy. A perfectlyinelastic collision is one in which the objects stick together:

Because the objects stick together, for a perfectly inelastic collision you need only oneequation to derive final from initial velocities.

Collisions in two dimensionsThe illustration below shows two objects colliding elastically within a two dimensional sys-tem. Prior to the collision, object 1 possesses all of the momentum in the system. The col-lision enables the transfer of some of this momentum to object 2. As can be seen in the

lower right corner of the illustration, the vector sum of the total linear momentum in the sys-

p1i p2i

p(1+2)f

+ =p1i p2i pi tot

p(1+2)f

=

beforecollision

aftercollision

total momentum remains the samebefore and after collision.

beforecollision

aftercollision

p1i

p1fp2f

p2i = 0

p2f

p1f p1i = ptot

Newton's second law in terms of momentumWe are all familiar with Newton's second law. An object's acceleration increases with theapplied force. The acceleration diminishes, however, as the same force is applied to amore massive object.

Momentum, p, is the product of the mass and the velocity. The rate of change of the veloc-ity is the acceleration. How might we express the rate of change of the momentum? Weknow that force equals the mass times the acceleration or, in other words, the mass timesthe rate of change of the velocity. It makes sense that another wave to say force is the rateof change of the momentum. To see this better, let's use calculus. Though it won't be onthe MCAT, calculus is always there.

Acceleration is the rate of change of the velocity with time, or you can say that accelerationis the first derivative of the velocity

Newton's second law can be restated with a derivative:

Because the momentum, p, is the product of the mass and the velocity, the mass times therate of change of the velocity equals the rate of change of the momentum:

Therefore, Newton's second law takes an even more simple form: The net force upon anobject equals the rate of change of the momentum:

ImpulseKeeping in mind that the force as the rate of change of momentum, lets explore the con-cept of impulse, which is the product of a force and the duration. We just showed thatforce is the rate of change of momentum. If you multiply a rate of change by a duration ofchange, you will get an amount of change. The impulse is the product of the force and thetime of action. It is the amount of momentum a certain force will produce over time.

a = dvdt

ΣF = mdvdt

mdvdt = dp

dt

ΣF = dpdt

© J.S Wetzel, 1993