5.5 The Lorentz Transformation

Show that if a time increment dt elapses for an observer who sees the particle moving with velocity v, it corresponds to a proper time particle increment for the particle of $d \tau = \gamma d t .$

Describe what happens to the angle $\alpha = \text{tan} \left(v / c\right) ,$ and therefore to the transformed axes in the figure, as the relative velocity v of the S and $\text{S} '$ frames of reference approaches c.

A man standing still at a train station watches two boys throwing a baseball in a moving train. Suppose the train is moving east with a constant speed of 20 m/s and one of the boys throws the ball with a speed of 5 m/s with respect to himself toward the other boy, who is 5 m west from him. What is the velocity of the ball as observed by the man on the station?

A man is running on a straight road perpendicular to a train track and away from the track at a speed of 12 m/s. The train is moving with a speed of 30 m/s with respect to the track. What is the speed of the man with respect to a passenger sitting at rest in the train?

In a frame at rest with respect to the billiard table, a billiard ball of mass m moving with speed v strikes another billiard ball of mass m at rest. The first ball comes to rest after the collision while the second ball takes off with speed v in the original direction of the motion of the first ball. This shows that momentum is conserved in this frame. (a) Now, describe the same collision from th…

In a frame S, two events are observed: event 1: a pion is created at rest at the origin and event 2: the pion disintegrates after time $\tau$. Another observer in a frame $\text{S} '$ is moving in the positive direction along the positive x-axis with a constant speed v and observes the same two events in his frame. The origins of the two frames coincide at $t = t ' = 0 .$ (a) Find the positions and timings of these two events…