5.9 Relativistic energy  (Page 10/16)

An observer sees two events $1.5\times {10}^{\mathrm{-8}}\text{s}$ apart at a separation of 800 m. How fast must a second observer be moving relative to the first to see the two events occur simultaneously?

An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a 500-m-long train simultaneously at the instant the middle of the train passes him at 50 m/s. Use the Lorentz transformation to find the time between the lightning strikes as measured by a passenger seated in the middle of the train.

Two astronomical events are observed from Earth to occur at a time of 1 s apart and a distance separation of $1.5\times {10}^9\text{m}$ from each other. (a) Determine whether separation of the two events is space like or time like. (b) State what this implies about whether it is consistent with special relativity for one event to have caused the other?

Two astronomical events are observed from Earth to occur at a time of 0.30 s apart and a distance separation of $2.0\times {10}^9\text{m}$ from each other. How fast must a spacecraft travel from the site of one event toward the other to make the events occur at the same time when measured in the frame of reference of the spacecraft?

A spacecraft starts from being at rest at the origin and accelerates at a constant rate g , as seen from Earth, taken to be an inertial frame, until it reaches a speed of c/ 2. (a) Show that the increment of proper time is related to the elapsed time in Earth’s frame by:

(b) Find an expression for the elapsed time to reach speed c/ 2 as seen in Earth’s frame. (c) Use the relationship in (a) to obtain a similar expression for the elapsed proper time to reach c /2 as seen in the spacecraft, and determine the ratio of the time seen from Earth with that on the spacecraft to reach the final speed.

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $12.0\times {10}^9\text{ly}$ away is receding from us at 0.900 c , at what velocity relative to us must we send an exploratory probe to approach the other galaxy at 0.990 c as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)

Suppose a spaceship heading straight toward the Earth at 0.750 c can shoot a canister at 0.500 c relative to the ship. (a) What is the velocity of the canister relative to Earth, if it is shot directly at Earth? (b) If it is shot directly away from Earth?