(a) John, who has a mass of m,, is initially at rest on a smooth surface when he strikes a solid cube of mass m, that is also initially at rest. He exerts a horizontal force of magnitude F on the cube for a period of time t = 0 to t = t,. Assume there is negligible friction between the cube and the surface, as well as negligible friction between John's shoes and the surface. Derive an equation for the final speed of the cube, v,, relative to the horizontal surface. Express your answer in terms of F, t;, and m. After time t,, John slides with a speed of v,. Derive an equation for the ratio of v /v, in terms of m; and m, and any physical constants if needed. (b) Assume m; >>> m. On the graph below, sketch a graph of both John's speed and the cube's speed as a function of time betweent=0and t = 2t,. Neither John nor the cube collide with anything after t,. On the vertical axis, label v, and v.. Label the graph lines with J and C respectively. o = Fmm e - (c) Assume the cube is now moving at a constant speed of v, toward a block, at rest, that has a mass m,. The cube and block collide, and then move together. The center of mass of the cube-block system has a constant speed of v, after the collision. i. Derive an equation for v, in terms of v; m,, m,, and physical constants, as appropriate. ii. Using the derived equation for v,, if the mass of the cube is much much greater than the mass of the block, what would be the approximate speed, v,, of the cube-block system? iii. Using the derived equation in i, if the mass of the cube is much much less than the mass of the block, what would be the approximate speed, v, of the cube-block system