6.03l Newton's law: oblique impacts

\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\) and \(\alpha + \beta = 90°\).

1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]

The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.

1. Find the value of \(\tan \alpha\). [5]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]

\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]

A smooth sphere \(A\) of mass \(m\) is projected with speed \(u\) along a smooth horizontal surface and strikes a stationary smooth sphere \(B\) of equal radius but of mass \(M\). The direction of motion of \(A\) before the impact makes an acute angle \(\theta\) with the line of centres at the moment of contact. After the impact, the direction of motion of \(A\) is perpendicular to the initial direction of motion of \(A\). The coefficient of restitution between the two spheres is \(e\). Given that \(Me \geq m\), prove that $$\tan^2 \theta = \frac{Me - m}{m + M}.$$ [8]