Notes on Relativity and Cosmology - Physics Department, UCSB

160 CHAPTER 6. DYNAMICS: ENERGY AND ...
North
South
Now, suppose that we have two particles that have the same rest mass m 0 ,
and which in fact are exactly the same when they are at rest. We will set
things up so that the two particles are moving at the same speed relative to the
room, but in opposite directions. We will also set things up so that they collide
exactly in the middle of the room, but are not moving exactly along either the
north-south axis or the east-west axis. Also, the particles will not quite collide
head-on, so that one scatters to each side after the collision. In the reference
frame of the room, the collision will look like this:
North
A before
B after
South
A after
B before
However, we will assume that the particles are nearly aligned with the east-west
axis and that the collision is nearly head-on, so that their velocities in the northsouth
direction are small. Note that I have labeled one of the particles ‘A’ and
one of them ‘B.’
To proceed, we will analyze the collision in a different reference frame. Suppose
that one of our friends (say, Alice) is moving rapidly to the east through the
room. If she travels at the right speed she will find that, before the collision and
relative to her, particle A does not move east or west but only moves north and
south. We wish to set things up so that the motion of particle A in Alice’s frame
of reference is slow enough that we can use the Newtonian formula p = mv for
this particle in this frame of reference. For symmetry purposes, we will have
another friend Bob who travels to the right fast enough that, relative to him,
particle B only moves in the north-south direction.
Now, suppose we set things up so that the collision is not only reversible, but
in fact looks exactly the same if we run it in reverse. That is, we suppose that
in Alice’s frame of reference, the collision looks like:
y
Before:
North
A
South
θ
B
x
B
After:
North
θ
South
A