Tensors and Relativity: Assignment 2
Problem
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Peter Dunsby Room 310a Applied Mathematics
- (1) A car 5 m long tries to get into a garage 3 m long by driving into it at a speed of
. Show that in the frame of the garage, the whole car can indeed enter the garage before its front strikes the wall. Also calculate the length of the garage as seen by the driver and prove that he expects to strike the wall 
seconds before the back of the car gets in through the back of the garage. Recalling that the maximum speed of propagation of information is c, explain how the car fits into the garage before the news that the front of the car has hit the garage wall reaches the back of the car.
- (2) In section 2.2 of the lecture notes we showed that the four- velocity of a particle in a frame
is given by
Show that for an observer at rest relative to
[ i.e. with a four- velocity 
], the particle's three- velocity 
can be written as
- (3) A particle of rest mass
moving with velocity 
along the x- axis collides elastically with a stationary particle of rest mass 
and as a result 
and 
are deflected through angles 
and 
respectively. If E and 
are the total energies of the particle 
before and after the collision respectively, show that
- (4) Define the four- velocity
and the four- acceleration 
of a particle in Special Relativity and specify how they relate to the Newtonian velocity 
and Newtonian acceleration 
. If the rest mass of the particle is 
, what is its four- momentum 
?
A particle moves with variable velocity
relative to some inertial frame, under the action of a force 
. Show that
where v is the magnitude of
. Infer that
if the acceleration is parallel to
, while
if the acceleration is perpendicular to
.Suppose the particle moves along the x- axis under a force of magnitude
being at rest at t=0. Show that the time taken to move to the point with coordinate
is
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