Using Centre of Mass to Help Solve Two Dimensional Momentum ...

Scattering in two dimensions
Let’s apply conservation of momentum and energy
to a non-trivial problem: the elastic scattering
of one body off another (elastic means no
energy is lost in the collision).
m
1
v 1 m 2
θ 1
θ 2
m
2
m
1
w
2
w
1
We will suppose body 2 (the target), is initially
at rest.
Conservation of energy
1
2 m 1 v 2
1 = 1
2 m 1 w 2
1 + 1
2 m 2 w 2
2 .
Conservation of momentum parallel to v 1
m 1 v 1 = m 1 w 1 cos θ 1 + m 2 w 2 cos θ 2 .
Conservation of momentum perpendicular to v 1
m 1 w 1 sin θ 1 = m 2 w 2 sin θ 2 .
Without loss of generality, we may take w 1
and
w 2
to be positive, and the scattering angles to lie
between 0 and π.
You may recall our earlier treatment of a collision
in one dimension. There, there were two
equations in two unknowns, the final speeds.
The final speeds were therefore fixed, once the
initial speeds were known.
Here, we have three equations in four unknowns,
the two final speeds and the two scattering
angles. Therefore, we won’t be able to solve for
them all. Let’s regard w 1
as a variable, and solve
for the other three in terms of w 1
.
Let’s solve for θ 1
(leaving θ 2
and w 2
as exercises).
We do this in several steps, which we
won’t reproduce here.
1) Eliminate θ 2
from the two momentum equations
by solving them for cos θ 2
and sin θ 2
, and
then using the identity cos 2 θ 2
+ sin 2 θ 2
= 1.
2) Eliminate w 2
by solving the energy equation
for w 2 2
and substituting it into the result of the
first step.
3) Solve the resulting equation for cos θ 1
. The result
is