Simple Nature - Light and Matter

b/A collision between two

pool balls is seen in two different

frames of reference. The solid

ball catches up with the striped

ball. Velocities are shown with

arrows. The second observer is

moving to the left at velocityu

compared to the first observer,

so all the velocities in the second

frame haveuadded onto them.

The two observers must agree on

conservation of energy.

As in the one-particle argument on page 89, the trick is to require
conservation of energy not just in one particular frame of reference,
but in every frame of reference. In a frame of reference moving at
velocityurelative to the first one, the velocities all haveuadded
onto them:^2

1

2

m 1 (v 1 i+u)^2 +

1

2

m 2 (v 2 i+u)^2 =

1

2

m 1 (v 1 f+u)^2 +

1

2

m 2 (v 2 f+u)^2

When we square a quantity like (v 1 i+u)^2 , we get the samev 12 ithat
occurred in the original frame of reference, plus twou-dependent
terms, 2v 1 iu+u^2. Subtracting the original conservation of energy
equation from the version in the new frame of reference, we have

m 1 v 1 iu+m 2 v 2 iu=m 1 v 1 fu+m 2 v 2 fu,

or, dividing byu,

m 1 v 1 i+m 2 v 2 i=m 1 v 1 f+m 2 v 2 f.

This is a statement that when you add upmvfor the whole system,
that total remains constant over time. In other words, this is a
conservation law. The quantitymvis calledmomentum, notatedp
for obscure historical reasons. Its units are kg·m/s.
Unlike kinetic energy, momentum depends on the direction of
motion, since the velocity is not squared. In one dimension, motion
in the same direction as the positivexaxis is represented with posi-
tive values ofvandp. Motion in the opposite direction has negative
vandp.
Jen Yu meets Iron Arm Lu example 1
.Initially, Jen Yu is at rest, and Iron Arm Lu is charging to the left,
toward her, at 5 m/s. Jen Yu’s mass is 50 kg, and Lu’s is 100 kg.
After the collision, the movie shows Jen Yu still at rest, and Lu
rebounding at 5 m/s to the right. Is this consistent with the laws
of physics, or would it be impossible in real life?
.This is perfectly consistent with conservation of mass (50 kg+100
kg=50 kg+100 kg), and also with conservation of energy, since
neither person’s kinetic energy changes, and there is therefore
no change in the total energy. (We don’t have to worry about
interaction energies, because the two points in time we’re con-
sidering are ones at which the two people aren’t interacting.) To
analyze whether the scene violates conservation of momentum,
we have to pick a coordinate system. Let’s define positive as

(^2) We can now see that the derivation would have been equally valid forUi 6 =
Uf. The two observers agree on the distance between the particles, so they
also agree on the interaction energies, even though they disagree on the kinetic
energies.
Section 3.1 Momentum in one dimension 133