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Momentum and Impulse
An object undergoing translational motion possesses linear momentum, p, a vector quantity, computed by taking the product of the mass and the velocity of the object.
p
The direction of the momentum vector is the same as that
of the velocity.
The Conservation of Momentum
Momentum is a useful concept. In any mechanical system composed of mutually interacting objects, though not subject to external forces, the net linear momentum remains
unchanged. This is the law of the conservation of momentum. The law has two mathematical forms describing the two types of collisions which objects within such a system can
have with each other, elastic or inelastic collisions.
In an elastic collision, the total translational kinetic energy, not only the momentum, of
the colliding objects remains unchanged before and after contact; none of the kinetic energy is transformed into other types of energy, such as heat or vibrational energy.
after
collision
p1i
p1f
p2i
+
p1i
p2f
p1f
+
p2i
=
pi tot
=
before
collision
total momentum remains the same
before and after collision.
p2f
=
pf tot
The mathematical expression to describe conservation of momentum is as follows:
Because also the kinetic energy is conserved in an elastic collision, the following expression also describes an elastic collision:
© J.S Wetzel, 1993
An inelastic collision is characterized by a decrease in the translational kinetic energy of
the objects upon collision. Some kinetic energy is transformed by the collision into other
forms of energy such as heat or the energy of sound waves. As with an elastic collision,
p1i
p2i
+
p1i
p2i
=
pi tot
=
before
collision
total momentum remains the same
before and after collision.
after
collision
p(1+2)f
p(1+2)f
momentum is conserved in inelastic collisions, though not kinetic energy. A perfectly
inelastic collision is one in which the objects stick together:
Because the objects stick together, for a perfectly inelastic collision you need only one
equation to derive final from initial velocities.
Collisions in two dimensions
The illustration below shows two objects colliding elastically within a two dimensional system. Prior to the collision, object 1 possesses all of the momentum in the system. The collision enables the transfer of some of this momentum to object 2. As can be seen in the
before
collision
p2f
after
collision
p1f
p2i = 0
p1i
p1f
p2f
p1i = ptot
lower right corner of the illustration, the vector sum of the total linear momentum in the sys-
Newton's second law in terms of momentum
We are all familiar with Newton's second law. An object's acceleration increases with the
applied force. The acceleration diminishes, however, as the same force is applied to a
more massive object.
Momentum, p, is the product of the mass and the velocity. The rate of change of the velocity is the acceleration. How might we express the rate of change of the momentum? We
know that force equals the mass times the acceleration or, in other words, the mass times
the rate of change of the velocity. It makes sense that another wave to say force is the rate
of change of the momentum. To see this better, let's use calculus. Though it won't be on
the MCAT, calculus is always there.
Acceleration is the rate of change of the velocity with time, or you can say that acceleration
is the first derivative of the velocity
a=
dv
dt
Newton's second law can be restated with a derivative:
ΣF = m dv
dt
Because the momentum, p, is the product of the mass and the velocity, the mass times the
rate of change of the velocity equals the rate of change of the momentum:
m
dv dp
=
dt dt
Therefore, Newton's second law takes an even more simple form: The net force upon an
object equals the rate of change of the momentum:
dp
ΣF = dt
Impulse
Keeping in mind that the force as the rate of change of momentum, lets explore the concept of impulse, which is the product of a force and the duration. We just showed that
force is the rate of change of momentum. If you multiply a rate of change by a duration of
change, you will get an amount of change. The impulse is the product of the force and the
time of action. It is the amount of momentum a certain force will produce over time.
© J.S Wetzel, 1993