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A  solid right circular cone, of mass $M$, has 
semi-vertical angle $\alpha$ and smooth surfaces. 
It stands with its base on
a smooth horizontal table. 
A particle of mass $m$ 
is projected 
 so that it strikes the curved surface of the cone
at speed $u$. 
The  coefficient of restitution
 between the particle  and the cone is $e$.
The impact has no rotational effect on the cone
and the cone has no vertical velocity
after the impact.
\begin{questionparts}
\item The particle strikes the cone in the direction of the normal at
  the point of impact. Explain why the trajectory of the particle
  immediately
after the impact is parallel to the normal to the surface of the cone.
Find an expression, in terms of $M$, $m$,
$\alpha$, $e$ and $u$, for the speed at which the cone slides
along the table immediately after impact.
\item
If instead the particle falls vertically onto the cone, show that the speed $w$
  at which the cone slides
along the table immediately after impact is given by
\[
w= \frac{mu(1+e)\sin\alpha\cos\alpha}{M+m\cos^2\alpha}\,.
\]
Show also that the value of $\alpha$ for which $w$ is greatest is given by
\[
\cos \alpha = \sqrt{ \frac{M}{2M+m}}\ .
\]
\end{questionparts}

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