Consider an inertial frame at rest with respect to the earth.An alien spaceship
ID: 1493889 • Letter: C
Question
Consider an inertial frame at rest with respect to the earth.An alien spaceship is observed to move along the x axis of thisframe in such a way that x(t) = [sin(t + /4) - b] / , where both x and t are measured in the inertial referenceframe, w = / 2 rad/h , and b = sin(/4). Assume alsothat the earth is located at the origin (x = 0) in thisframe. a) Argue that the ship passes the earth at t = 0 and again att = 1.0h. (Hint: The value of t is /2 at thistime.) b) Draw a quantitatively accurate spacetime diagram of thespaceship's worldline, labeling the events where and when it passesthe earth as events A and B. c) Show that the ship's x-velocity is vx=cos(t + /4) as measured in the inertial frame attachedto the earth. (Hint: You don't need to know anything aboutrelativity!) d) Find the proper time measured by clocks on the ship betweenthe time the ship passes earth the first time and the time itpasses the second time. (Hint: 1 - cos2 x =sin2 x)
Explanation / Answer
a) x(t) = [sin(wt+pi/4)-b]/w
x(t) = [sin(pit/2 + pi/4)-sin(pi/4)]/w
x(0) = 0
x(1) = sin(3pi/4) - sin(pi/4) = 0
so the ship passes when x = 0
b) Everything for the space ship is the same except that everything is shifted by phi = 180 degrees
c) v = dx/dt = cos(wt+pi/4)w/w = cos(wt+pi/4)
d) time = t
(sin(pit/2 + pi/4)-sin(pi/4))/w - (sin(pi*2t/2 + pi/4)-sin(pi/4))/w = 1 hour