Standard +0.8 This is a two-stage collision problem requiring conservation of momentum, Newton's restitution law, and careful analysis of conditions for a second collision. Part (a) is a standard 'show that' requiring systematic application of collision equations. Part (b) requires determining when particle velocities allow a second collision after Q rebounds from the wall, involving algebraic manipulation and inequality reasoning—more demanding than typical A-level mechanics but standard for FM1.
Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of \(P\) is \(3 m\) and the mass of \(Q\) is \(4 m\).
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 7 } ( 9 e + 2 )\)
After the collision with \(P\), particle \(Q\) collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(Q\).
The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 2 }\)
Find the complete range of possible values of \(e\) for which there is a second collision between \(P\) and \(Q\).
\begin{enumerate}
\item Two particles, $P$ and $Q$, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.\\
The mass of $P$ is $3 m$ and the mass of $Q$ is $4 m$.\\
Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $u$.\\
The coefficient of restitution between $P$ and $Q$ is $e$.\\
(a) Show that the speed of $Q$ immediately after the collision is $\frac { u } { 7 } ( 9 e + 2 )$
\end{enumerate}
After the collision with $P$, particle $Q$ collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of $Q$.\\
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 2 }$\\
(b) Find the complete range of possible values of $e$ for which there is a second collision between $P$ and $Q$.
\hfill \mbox{\textit{Edexcel FM1 2022 Q5 [10]}}