Figure 37-16 shows a ship (attached to reference frame ) passing us (standing in reference frame ) with velocity . A proton is fired at speed relative to the ship from the front of the ship to the rear. The proper length of the ship is . What is the temporal separation between the time the proton is fired and the time it hits the rear wall of the ship according to (a) a passenger in the ship and (b) us? Suppose that, instead, the proton is fired from the rear to the front. What then is the temporal separation between the time it is fired and the time it hits the front wall according to (c) the passenger and (d) us?

Question: The mass of a muon is 207 times the electron mass; the average lifetime of muons at rest is . In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of . For the moving muons, what are (a) , (b) K, and (c) p (in MeV/c)?

Sam leaves Venus in a spaceship headed to Mars and passes Sally, who is on Earth, with a relative speed of . (a) Each measures the Venus–Mars voyage time. Who measures a proper time: Sam, Sally, or neither? (b) On the way, Sam sends a pulse of light to Mars. Each measures the travel time of the pulse. Who measures a proper time: Sam, Sally, or neither?

(Come) back to the future. Suppose that a father is 20.00 y older than his daughter. He wants to travel outward from Earth for 2.000 y and then back for another 2.000 y (both intervals as he measures them) such that he is then 20.00 y younger than his daughter. What constant speed parameter (relative to Earth) is required?

Quite apart from effects due to Earth’s rotational and orbital motions, a laboratory reference frame is not strictly an inertial frame because a particle at rest there will not, in general, remain at rest; it will fall. Often, however, events happen so quickly that we can ignore the gravitational acceleration and treat the frame as inertial. Consider, for example, an electron of speed v =0.992c, projected horizontally into a laboratory test chamber and moving through a distance of 20 cm. (a) How long would that take, and (b) how far would the electron fall during this interval? (c) What can you conclude about the suitability of the laboratory as an inertial frame in this case?

How much work must be done to increase the speed of an electron from rest to (a) 0.500c, (b) 0.990c, and (c) 0.9990c?