Momentum：練習問題

Two smooth spheres \(P\) and \(Q\) of masses \(3m\) and \(m\) respectively are moving on a smooth horizontal table with velocities \(2u\) and \(u\) in the same direction. They collide directly. If the collision is perfectly inelastic, find the common velocity of the spheres after the impact.

A smooth sphere \(S\) moves with speed \(V\) and strikes an identical smooth sphere \(T\) which is at rest. The direction of motion of \(S\) makes an angle of \(60^\circ\) with the line of centers at the moment of impact. Given the collision is perfectly elastic (\(e=1\)), find the angle that the velocity of \(S\) makes with the line of centers after the impact.

Two smooth spheres \(A\) and \(B\) of equal radius and masses \(2m\) and \(m\) respectively are moving towards each other in a straight line with speeds \(u\) and \(2u\). The collision is perfectly elastic. What is the speed of sphere \(B\) after the collision?

A smooth sphere of mass \(m\) hits a fixed vertical wall directly with speed \(u\). If the kinetic energy lost during the impact is \(\frac{3}{4}\) of the initial kinetic energy, find the coefficient of restitution \(e\).

A smooth sphere of mass \(m\) moving with speed \(u\) on a horizontal floor strikes a fixed vertical wall at an angle of \(30^\circ\) to the wall. The coefficient of restitution between the sphere and the wall is \(0.5\). Find the speed of the sphere after the impact.

A smooth sphere of mass \(m\) moving with speed \(u\) collides directly with a stationary smooth sphere of mass \(km\). The coefficient of restitution between the spheres is \(e\). Show that the speed of the sphere of mass \(m\) after the collision is \(u\left|\frac{1-ke}{1+k}\right|\).

Two smooth spheres \(P\) and \(Q\) of equal radius and masses \(m\) and \(2m\) respectively are moving on a smooth horizontal floor. \(P\) moves with speed \(u\) and strikes \(Q\), which is at rest. The impact is oblique such that the velocity of \(P\) before impact makes an angle of \(30^\circ\) with the line of centers. If the coefficient of restitution is \(0.5\), find the magnitude of the velocity of \(Q\) after the impact.

A smooth sphere \(A\) moving with speed \(u\) collides with a fixed smooth vertical wall. The direction of motion of \(A\) before the impact makes an angle \(\alpha\) with the wall. Given that the coefficient of restitution is \(e\), find an expression for the kinetic energy of \(A\) after the impact as a fraction of its initial kinetic energy.

A smooth sphere A of mass \(2m\) is moving with speed \(u\) on a smooth horizontal plane. It collides directly with another smooth sphere B of mass \(m\), which is at rest. The coefficient of restitution between the spheres is \(e\).

(a) Find the speeds of A and B immediately after the collision in terms of \(u\) and \(e\).

(b) If the collision is perfectly elastic (\(e=1\)), determine the percentage change in speed of sphere A.