(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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