Answers to Chapter 4 Questions

4.1) We want to find the equation of the straight line that corresponds to the X_bus-axis (T_bus = 0 line) on the Anne and Bev spacetime diagram. The general equation for any straight line on an Anne and Bev spacetime diagram is,

T_Earth = mX_Earth + b.

The X_bus-axis passes through the origin where Chuck’s watch read zero, T_Earth = 0 and X_Earth = 0, and the point where Ed’s watch read zero which, according to Anne and Bev, has coordinates T_Earth = vD_Earth/c² and X_Earth = D_Earth. Using the coordinates for those two points, we get two equations,

0 = m0 + b → b = 0

vD_Earth/c² = mD_Earth → m = v/c².

4.2a) The shrinkage factor for v = 0.8 ft/ns equals 0.6.

4.2b) Anne and Bev measure the 100 foot bus to be only 60 feet long.

4.2c) 60 feet

4.2d) The worldline of the light flash is given by X_flash = T_Earth. Dean’s worldline is given by X_Dean = 60 + 0.8T_Earth. The two worldlines meet when X_flash = X_Dean and that is the spot where Bev needs to be standing to be adjacent to Dean when the flash arrives. The flash reaches Dean when T_Earth = 60 + 0.8T_Earth. Solving for T_Earth, we get 60/0.2 = 300 ns. In 300 ns, the flash traveled 300 feet, therefore Bev needs to be 300 feet to the right of Anne.

4.3) Since a mile is 5280 feet long, it takes Dean 528 seconds to walk a mile. This 8 minutes and 48 seconds.

4.4) I assume you calculator did not get an answer different than 1, but if it did, it will agree “exactly” with the approximate value of 1 – x/2!