Chapter 6, Problem 47P

** You are investigating a newly discovered particle X that has an unknown mass M and moves with a constant but unknown speed u. In your experiments you observe collisions between particle X and different test particles with known masses m that are initially at rest. You let the particles collide head-on and stick together, and you measure the speed v of the combined-particle object. From other observations you know that in all experiments the objects move along the same straight line before and after the collision. Your data are shown in the table at right. Note that the masses are expressed in units of reference mass $m_0$ and the speed in units of reference speed $v_0$.

m (m0)	v(v0)
2.0	0.55
4.0	0.45
6.0	0.35
8.0	0.30
10.0	0.25

(a) Derive an expression for v in terms of M, m, and u. Indicate any assumptions that you made. (b) Determine M and u using the data in the table. (Hint: Rearrange the expression that you derived in (a) to obtain a new equation with the linear dependence on m.)