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Collisions
Objective: Describe what is meant by the a “collision”; state the principle of conservation of
momentum and relate it to the momentum principle; state assumptions made for a collision
when the interaction forces between particles are much larger than external forces; apply
conservation of momentum to a head-on collision.
Review
Suppose you throw a superball at a wall and it rebounds with the same speed as it had just before the
collision (this is not practical since all real balls will lose kinetic energy during the collision with the wall).
What is the direction of the change in momentum of the ball?
What is the direction of the net force on the ball (which is equal in this case to the force of the wall on
the ball)?
But what if we consider the ball and earth as a system? Then, in this case, the net external force on the
system during the collision is zero and the total momentum is constant. Thus, the change in the momentum
of the earth must be equal in magnitude and opposite direction the change in the momentum of the ball. As
a result, the change in the total momentum of the system is zero, and the total momentum of the system is
constant.
Conservation of momentum
According to the momentum principle, if the net external force on a system is zero, the change in the
momentum of the system is zero, thus the momentum of the system is constant. Alternatively, this can be
stated as
∆P~system = −∆P~surroundings
(1)
If there are no impulses (F~ ∆t) on the system from the surroundings, then the momentum of the system
is constant. This principle is very useful for analyzing collisions. In a collision, two or more particles interact
for a very short time, and during that time, the forces of the particles on each other are much greater than
the net external force on the system. Therefore, we can approximate the total momentum as being conserved
during the collision.
This approximation is useful for analyzing car accidents to determine initial velocities at impact, etc.
However, physicists use collisions as the primary method of understanding what matter is made of. It was the
Rutherford experiment that showed how tiny the nucleus is compared to the atom. In modern experiments,
physicists collide high-kinetic energy protons (and other particles) to explore the kinds of particles that are
generated during the collision. This is how we determine the masses of quarks, for example.
An elastic collision is a collision where the internal energy of the particles in the system does not change.,
∆Eint = 0.
A head-on elastic collision
As a detailed example, let’s consider a head-on elastic collision in the x-direction between an alpha particle
(He nucleus) and a gold nucleus where the gold nucleus is initially at rest and the alpha particle has an
initial momentum.
Applying conservation of momentum gives:
p1x = p3x + p4x
(2)
Applying conservation of energy gives:
p21x
p2
p2
= 3x + 4x
2m
2m 2M
The result of solving these two equations for the two unknowns, p3x and p4x gives
m−M
p3x =
p1x
m+M
(3)
(4)
and
p4x =
2M
M +m
p1x
(5)
What if you turn it around? Instead of shooting the lightweight alpha particle at the heavy gold nucleus,
shoot the gold nucleus at the alpha particle? What do you think will happen? We can just switch M and
m in the above equations and note that 3 now refers to the gold nucleus and 4 refers to the alpha particle;
therefore,
M −m
p3x =
p1x ≈ p1x
(6)
M +m
and
p4x =
2m
m+M
p1x ≈
2m
p1x
M
(7)
Note that the momentum of the gold nucleus hardly changes at all. However, the alpha particle will have
a final x-velocity that is 2 times greater than the initial x-velocity of the gold nucleus.
Application
1. Two Hot-Wheels cars sit on a low-friction track. A compressed spring is placed in between the cars
and, at some instant, is released. The cars have equal masses of 0.020 kg.
(a) What is the momentum of the system (defined to include both cars) before the spring is released?
(b) According to Newton’s second law, what can you say about the momentum of the system before
and after the collision?
(c) What is the momentum of the system after the spring is released?
(d) Suppose one car is moving with a speed of 0.75 m/s in the +x direction after the spring is released.
What is the velocity of the other car after the spring is released?
(e) If you repeat the experiment with two cars and a different spring, one with a mass of 0.020 kg
and the other with a mass of 0.030 kg, and find that the heavier car is moving with a speed of
0.50 m/s in the +x direction after the spring is released. What is the velocity of the lighter car
after the spring is released?
2. An alpha particle of mass 4 g/mol collides head-on with an oxygen nucleus of mass 16 g/mol that is
at rest. Before the collision, the alpha particle’s velocity is 4.0 × 105 m/s in the +x direction.
(a) Use the momentum principle to argue why you believe that the momentum of the system (of the
alpha particle and oxygen nucleus) is constant?
(b) If after the collision, the oxygen nucleus moves with a velocity of 1.6×105 m/s in the +x direction,
what is the magnitude and direction of the velocity of the alpha particle after the collision?
(c) If this collision takes place during a time interval of 1 × 10−12 s, what is the magnitude of the
average net force on the alpha particle during the collision?
(d) According to the principle of reciprocity, what can you say about the force of the alpha particle
on the oxygen nucleus compared to the force of the oxygen nucleus on the alpha particle?