Momentum：練習問題

Two smooth spheres \( A \) and \( B \) of masses \( m \) and \( 3m \) respectively are moving towards each other with speeds \( 2u \) and \( u \) on a smooth horizontal table. The impact is direct and the coefficient of restitution is \( e \). Find the range of values of \( e \) such that the direction of motion of sphere \( A \) is reversed.

A smooth sphere \( A \) of mass \( m \) moving with speed \( u \) collides obliquely with an identical smooth sphere \( B \) of mass \( m \) which is at rest. The direction of motion of \( A \) before impact makes an angle of \( 60^\circ \) with the line of centers. If the coefficient of restitution is \( e = 1 \), find the angle between the velocities of the two spheres after the impact.

A smooth sphere \( A \) of mass \( m \) moves with speed \( u \) and strikes a smooth sphere \( B \) of mass \( km \) which is at rest. The impact is direct and the coefficient of restitution is \( e \). If sphere \( A \) is brought to rest by the impact, find the value of \( k \) in terms of \( e \).

A smooth sphere strikes a fixed smooth horizontal plane with speed \( u \) at an angle \( \alpha \) to the normal. The coefficient of restitution between the sphere and the plane is \( e \). The sphere loses exactly half of its kinetic energy during the impact. Find an expression for \( e^2 \) in terms of \( \alpha \).

A smooth sphere of mass \( m \) moving with speed \( u \) in a direction making an angle \( \alpha \) with the normal to a smooth fixed vertical wall strikes the wall. The coefficient of restitution between the sphere and the wall is \( e \). Given that the sphere loses \( \frac{3}{4} \) of its kinetic energy during the impact, and \( e = \frac{1}{3} \), find the value of \( \tan^2 \alpha \).

A small smooth sphere \(A\) moving with speed \(u\) on a horizontal table collides with an identical sphere \(B\) which is at rest. The direction of motion of \(A\) before impact makes an angle of \(45^\circ\) with the line of centers. Given that the coefficient of restitution is \(e = \frac{1}{3}\), find the angle between the velocity of \(A\) and the line of centers after the impact.

A smooth sphere of mass \(m\) moving with speed \(u\) collides obliquely with a fixed smooth vertical wall. The direction of motion of the sphere before impact makes an angle of \(30^\circ\) with the wall. Given the coefficient of restitution is \(e = 0.5\), find the speed of the sphere after the impact in terms of \(u\).