A space probe, initially at rest, undergoes an internal mechanical malfunction and breaks into three pieces. One piece of mass m 1 5 48.0 kg travels in the positive x - direction at 12.0 m/s, and a second piece of mass m 2 5 62.0 kg travels in the xy-plane at an angle of 105° at 15.0 m/s. The third piece has mass m 3 5 112 kg. (a) Sketch a diagram of the situation, labeling the different masses and their velocities. (b) Write the general expression for conservation of momentum in the
x- and y-directions in terms of m 1, m 2, m 3, v 1, v 2, and v 3 and the sines and cosines of the angles, taking u to be the unknown angle. (c) Calculate the final x-components of the momenta of m 1 and m 2. (d) Calculate the final y-components of the momenta of m 1 and m 2. (e) Substitute the known momentum components into the general equations of momentum for the x- and y-directions, along with the known mass m 3. (f) Solve the two momentum equations for v 3 cos u and v 3 sin u, respectively, and use the identity cos2 u 1 sin2 u 5 1 to obtain v 3. (g) Divide the equation for v 3 sin u by that for v 3 cos u to obtain tan u, then obtain the angle by taking the inverse tangent of both sides. (h) In general, would three such pieces necessarily have to move in the same plane? Why?