Challenging +1.2 This is a standard M4 oblique collision problem requiring resolution of velocities parallel and perpendicular to the line of centres, application of Newton's experimental law, and conservation of momentum. While it involves multiple steps and algebraic manipulation to reach the given result, the technique is well-practiced in M4 and follows a predictable structure with no novel insight required. The 11 marks reflect the working length rather than conceptual difficulty.
\includegraphics{figure_1}
A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).
Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]
\includegraphics{figure_1}
A smooth sphere $P$ lies at rest on a smooth horizontal plane. A second identical sphere $Q$, moving on the plane, collides with the sphere $P$. Immediately before the collision the direction of motion of $Q$ makes an angle $\alpha$ with the line joining the centres of the spheres. Immediately after the collision the direction of motion of $Q$ makes an angle $\beta$ with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is $e$.
Show that $(1-e) \tan \beta = 2 \tan \alpha$. [11]
\hfill \mbox{\textit{Edexcel M4 2005 Q3 [11]}}