Question

A spaceship travels through the dock of a space station without slowing down. The speed of the spaceship relative to the station is v = 4c/5, where c is the speed of light. Consider frames of reference in the standard configuration with v and all distances aligned along the x-axis. Primed variables refer to events in the station frame and unprimed to events in the spaceship frame. The dock has a length of L′ = 200 m in the station frame.

(a) (i) In the station frame, calculate the time, T′, that it takes for the front of the spaceship to travel the length of the dock.

(ii) Define two events in the spaceship frame at (t = 0, x = 0) and (t = T, x = 0) and use appropriate Lorentz transforms to transform them to the station frame. These events correspond to the front of spaceship passing the entrance of the dock and the exit of the dock, respectively.

(iii) In the spaceship frame, calculate the time, T , that it takes for spaceship to travel the length of the dock.

(b) A fuel transfer will be attempted while the spaceship passes through the dock. The spaceship length is L = 289 m in the spaceship frame. In order for the transfer to be successful, force fields must be closed simultaneously in the station frame at each end of the dock to enclose the entire spaceship for a short period of time.

(i) In the station frame, the event corresponding to closing the front of the dock is at (t′ = 0, x′ = 0) and the event corresponding to closing the rear of the dock is at
(t′ = 0, x′ = L′). Perform the Lorentz transformation on each of these coordinates to determine the coordinates t,x in the spaceship frame in terms of L′, v and c.

(ii) Calculate the difference between the spatial coordinates in the spaceship frame. On this basis, comment on whether the phenomenon of length contraction will allow the entire spaceship to fit into the dock.

Answer 1