A collision occurs when two or more objects
hit each other. When objects collide, each object feels a force
for a short amount of time. This force imparts an impulse, or changes the
momentum of each of the colliding objects. But if the system of
particles is isolated, we know that momentum is conserved. Therefore,
while the momentum of each individual particle involved in the collision
changes, the total momentum of the system remains constant.
The procedure for analyzing a collision depends on whether
the process is elastic or inelastic. Kinetic
energy is conserved in elastic collisions, whereas kinetic energy
is converted into other forms of energy during an inelastic collision.
In both types of collisions, momentum is conserved.
Anyone who plays pool has observed elastic collisions.
In fact, perhaps you’d better head over to the pool hall right now
and start studying! Some kinetic energy is converted into sound
energy when pool balls collide—otherwise, the collision would be
silent—and a very small amount of kinetic energy is lost to friction.
However, the dissipated energy is such a small fraction of the ball’s
kinetic energy that we can treat the collision as elastic.
Equations for Kinetic Energy and Linear Momentum
Let’s examine an elastic collision between two particles
, respectively. Assume
that the collision is head-on, so we are dealing with only one dimension—you
are unlikely to find two-dimensional collisions of any complexity
on SAT II Physics. The velocities of the particles before the elastic
, respectively. The velocities
of the particles after the elastic collision are
. Applying the law of
conservation of kinetic energy, we find:
Applying the law of conservation of linear momentum:
These two equations put together will help you solve any
problem involving elastic collisions. Usually, you will be given
, and can then manipulate the two equations
to solve for
pool player hits the eight ball, which is initially at rest, head-on
with the cue ball. Both of these balls have the same mass, and the
velocity of the cue ball is initially . What are the velocities of the two balls after
the collision? Assume the collision is perfectly elastic.
into the equation for conservation of kinetic energy
Applying the same substitutions to the equation
for conservation of momentum, we find:
If we square this second equation, we get:
By subtracting the equation for kinetic energy from this
equation, we get:
The only way to account for this result is to conclude
. In plain English, the cue ball and the
eight ball swap velocities: after the balls collide, the cue ball
stops and the eight ball shoots forward with the initial velocity
of the cue ball. This is the simplest form of an elastic collision,
and also the most likely to be tested on SAT II Physics.
Most collisions are inelastic because kinetic energy is
transferred to other forms of energy—such as thermal energy, potential
energy, and sound—during the collision process. If you are asked
to determine if a collision is elastic or inelastic, calculate the
kinetic energy of the bodies before and after the collision. If
kinetic energy is not conserved, then the collision is inelastic.
Momentum is conserved in all inelastic collisions.
On the whole, inelastic collisions will only appear on
SAT II Physics qualitatively. You may be asked to identify a collision
as inelastic, but you won’t be expected to calculate the resulting
velocities of the objects involved in the collision. The one exception
to this rule is in the case of completely inelastic collisions.
Completely Inelastic Collisions
A completely inelastic collision, also called a “perfectly”
or “totally” inelastic collision, is one in which the colliding
objects stick together upon impact. As a result, the velocity of the
two colliding objects is the same after they collide. Because
, it is possible to solve problems asking
about the resulting velocities of objects in a completely inelastic
collision using only the law of conservation of momentum.
gumballs, of mass m and mass 2m respectively,
collide head-on. Before impact, the gumball of mass m is
moving with a velocity , and the gumball of mass 2m is
stationary. What is the final velocity, , of the gumball wad?
First, note that the gumball wad has a mass of m
The law of conservation of momentum tells us that
, and so
. Therefore, the final gumball wad
moves in the same direction as the first gumball, but with one-third
of its velocity.
Collisions in Two Dimensions
Two-dimensional collisions, while a little more involved
than the one-dimensional examples we’ve looked at so far, can be
treated in exactly the same way as their one-dimensional counterparts.
Momentum is still conserved, as is kinetic energy in the case of
elastic collisions. The significant difference is that you will
have to break the trajectories of objects down into x-
and y-components. You will then be able to deal
with the two components separately: momentum is conserved in the x direction,
and momentum is conserved in the y direction. Solving
a problem of two-dimensional collision is effectively the same thing
as solving two problems of one-dimensional collision.
Because SAT II Physics generally steers clear of making
you do too much math, it’s unlikely that you’ll be faced
with a problem where you need to calculate the final velocities
of two objects that collide two-dimensionally. However, questions
that test your understanding of two-dimensional collisions qualitatively
are perfectly fair game.
pool player hits the eight ball with the cue ball, as illustrated
above. Both of the billiard balls have the same mass, and the eight
ball is initially at rest. Which of the figures below illustrates
a possible trajectory of the balls, given that the collision is
elastic and both balls move at the same speed?
The correct answer choice is D,
because momentum is not conserved in any of the other figures. Note
that the initial momentum in the y direction is
zero, so the momentum of the balls in the y direction
after the collision must also be zero. This is only true for choices D and E.
We also know that the initial momentum in the x direction
is positive, so the final momentum in the x direction
must also be positive, which is not true for E.