Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

• We saw that the relativistic momentum p = γmu of a mass m can also be expressed as

where dt0 denotes the proper time between two neighboring points on the body’s path and has the same value for all observers. Show that the relativistic energy E = γmc2 can similarly be rewritten as

The relations (2.60) and (2.61) make it easy to see how p and E transform from one inertial frame to another. (See Problem 1.)

Problem 1

••• (a) Suppose that a mass m has momentum p and energy E, as measured in a frame S. Use the relations (2.60) and (2.61) and the known transformation of dr and dt to find the values of p′ and E′ as measured in a second frame S′ traveling with speed υ along Ox. (Notice that apart from some factors of c, the quantities p and E transform just like r and t. Remember that dt0 has the same value for all observers.) (b) Use the results of part (a) to prove the following important result: If the total momentum and energy of a system are conserved as measured in one inertial frame S, the same is true in any other inertial frame S′.