Irodov Problem 1.361

In this problem we can use the solution we found in Problem 1.358, namely








Let us find the velocity of the particle moving along the y-axis relative to the particle moving on the x-axis. For this, in our above equations we have to substitute, V= v1 , vx=0, vy=v2. Thus, we have, the relative velocity is given by,

Irodov Problem 1.360

For this problem we can use the solution found in Problem 1.359 namely,





We can determine the velocity of one of the rods relative to the other by setting v1=v2=v to obtain,





To one rod thus, the other rod would appear to be shrunk and of length,

Irodov Problem 1.359

a) The position of the particles as seen in the laboratory frame of reference are given by x1 = v1t and x2 = -v2t. The relative velocity is thus simple the time derivative of x1-x2 given by v1+v2.

b) For this part of the problem we could use the result obtained from Problem 1.358 namely,





To obtain the velocity of particle 2 as seen from particle 1 we have to substitute V=v1 and vx = -v2 in the above equation. Hence, we obtain,

Irodov Problem 1.358

The position of the particle in frame K as a function of time is given by,





Using Lorentz transform we can determine the position of the particle as a function of time t as seen in frame K' as,








The velocity of the particle as seen in frame K' is then given by,

Irodov Problem 1.357

In Newton's view of space no matter which inertial frame we look from distance between two points in space remains the same. In other words if we have two frames K and K' with constant relative velocity w.r.t each other then .

In Einstein's world however, space shrinks as seen from the eyes of a moving observer, thus, clearly the distance between two points is not an invariant between two inertial frames. Further, time also dilates or expands for the moving observer.

It turns out that the corresponding invariant in Einstein's space-time is given by



This invariant property of Einstein's space-time can be applied to solved the problem. Let us call this invariant as the space-time distance between two events.

a) The space-time distance between events A and B as seen from the figure is given by . The space-time distance between events A and B as seen from any inertial will be the same. Thus, in a frame where the two events occur at the same point in space, we have,





b) Same as above the space-time distance between events A and C is given by
.

In a frame where the events occur at the same time, the distance between them will be given by

Irodov Problem 1.356

This is best solved and explained through an example. Say a person located at x=0, fires a gun at time t=0 with respect to frame K. Say the bullet takes time in frame, travels a distance d along the positive x-axis and then hits a board. In other words, the space-time coordinates of the event corresponding to the man firing the gun are (0,0) and that of the event corresponding to the bullet hitting the board are (,d). Now consider a frame K' which is moving at a velocity v along the x-axis. As seen from frame K', using the Lorentz transform, the space-time coordinates of the event of bullet hitting the board are given by . In order for the causality to be disturbed, it should appear to the observer in K' that the bullet hit the board before it was shot. In other words,








Now suppose that the bullet traveled at a velocity vb. Then we have . From equation (1) we have the condition for causality to be contradicted as,



What equation (2) tells us is that in order for causality to be contradicted, at least one among the bullet or the frame K' must be traveling faster that the speed of light! Since nothing can travel faster than the speed of light, causality could never be contradicted.