here - Department of Physics, HKU

CHAPTER 1. PRE-SPECIAL RELATIVITY PHYSICS 2
the gravitational attraction of the Earth pulling us down, at the same time,
there is a force of the same strength pulling the Earth “up.”
Momentum, or linear momentum, of a particle is defined as the product
Hence, the Newton’s second law can be re-stated as
p = mv . (1.2)
F = dp
dt . (1.3)
This is the form that will be generalized to special relativity. We found that
in the absence of any external force, by Eq. (1.1), the total momentum of a
system remains constant. This is the conservation of momentum.
The addition of velocities in classical mechanics is very simple. For example,
if a train is moving with velocity v t relative to the station and a ball
is moving with velocity v b relative to the train, then relative to the station,
the ball is moving with velocity v t + v b .
The kinetic energy of a particle is given by
K. E. = 1 2 mv2 = p2
2m
(1.4)
where v and p are the magnitude of velocity and momentum respectively. If
we want to change the velocity of the particle, a force must act on it. The
change of kinetic energy is equal to the work done W, which, for constant
force, is the dot product of the force vector F and the displacement vector r
of the particle
W = F · r . (1.5)
If the force is not constant, consider the infinitesimal small displacement
dr, which means, consider the particle moves from position r to r + dr, the
contribution to the work done is F · dr. The total work done is to integrate
W =
∫ r1
r 0
F · dr (1.6)
where r 0 and r 1 are the initial and final points.
Apart from the kinetic energy, there are other kinds of energies, like the
potential energy, chemical energy or nuclear energy. If we sum up all kinds of
energies in an isolated system, the total energy also remains constant. This
is the principle of conservation of energy.
1.2 Galilean Transformations
We will discuss Galilean transformations after we talk about two important
concepts.