Question

Far out in space, a bored astronaut fires an antique gun at a sack of rocks (mass M). The bullet (mass m) bounces around inside the sack a few times, eventually emerging with a reduced, but still high, speed. As the bullet flies off into space, the astronaut remembers with dismay that it was a priceless historical artifact. Fortunately, the astronaut knows the initial speed v_0 of the bullet, and that the sack was initially at rest but after being hit was found to be moving with speed v_s at angle theta_s as shown in the diagram. In what direction should the astronaut pursue the bullet, assuming that he starts from the initial location of the sack, and what minimum speed v_b must he go faster than in order to catch up with it? Use the numerical values given below, and enter the direction as an angle theta_b measured clockwise from the x-axis.

Answer 1

"According to the conservation of momentum, for the system p _f = p _i mj V _b^- > + m V _s^- > = m V _0^- > m V _b^- > = m V _0^- > - m V _s^- > V _b^- > = V _0^- > - (M/m)V _s^- > = (1550 m/s)i^- (45/0.025)(0.315) (cos theta i^+ sin theta j^) V _b^- > = (1550 m/s)i^- 567(cos 29.5 i^+ sin 29.5 j^] V _b^- > = (1550 m/s) i^- 493.5 i^- 279.2 j^V _b^- > = (1056.5 m/s) i^- (279.2 m/s)j^so its magnitude is