CMI CM-I Problem Sheet 9 (2012)

41. Suppose the frame $S'$ has a velocity $\vec{v}\ =\ v_1\,\hat{\epsilon}_1\, +\,v_2\ \hat{\epsilon}_2\,+\, v_3\hat{\epsilon}_3$ with respect to the frame $S$. Find the Lorentz transformation relating an event between the two frames.

42. A ball is bouncing back and forth between two walls [ $x_1\,=\,0$ and $x_1\,=\,L$] with speed $u$ along the $x_1$ direction. What will be time taken by the ball to oscillate between the walls as viewed from a frame $S'$ which

1. has a velocity $v\,\hat{\epsilon}_1$?
2. has a velocity $v\,\hat{\epsilon}_2$?

43. There are two clocks $A\, ,\, B$ located at $(-L/2,0,0)$ and $(L/2,0,0)$ respectively. There is a mirror $M$ parallel to the $x_1-x_3$ plane at $x_2\,=\,L_1$.The clock $B$ is synchronised with $A$ by sending a light beam from $M$ and reaching $B$. If the signal was sent at $t\,=\,0$ from $A$, then the clock is adjusted to $t\ =\ [2(L^2/4\, + \,L_1^2)^{1/2}]/c$. What will be the time lag as  viewed from a frame $S'$ which is moving with $v\hat{\epsilon}_1$?

44. Suppose two events as seen from $S$ have co-ordinates $(x_0(A),x_1(A),0,0)$ and $(x_0(B),x_1(B),0,0) $ and are space like with respect to each other. Find the frame in which both the events occur at the same time. ( In particular find the velocity of the frame $S'$ in which they occur simultaneously. It should be less than $c$ !)

45. A particle decays with a life time of $T$ in it's rest frame. Suppose the particle is moving and travelled a distance $D$ before decaying. Find its speed.