2002 Paper 3 Q11

Year: 2002
Paper: 3
Question Number: 11

Course: UFM Mechanics
Section: Momentum and Collisions 2

Difficulty: 1700.0 Banger: 1484.0
momentum and collisions coefficient of restitution repeated impacts geometric series convergence triangular surface oblique impact angle of incidence

Problem

A particle moves on a smooth triangular horizontal surface \(AOB\) with angle \(AOB = 30^\circ\). The surface is bounded by two vertical walls \(OA\) and \(OB\) and the coefficient of restitution between the particle and the walls is \(e\), where \(e < 1\). The particle, which is initially at point \(P\) on the surface and moving with velocity \(u_1\), strikes the wall \(OA\) at \(M_1\), with angle \(PM_1A = \theta\), and rebounds, with velocity \(v_1\), to strike the wall \(OB\) at \(N_1\), with angle \(M_1N_1B = \theta\). Find \(e\) and \(\displaystyle {v_1 \over u_1}\) in terms of \(\theta\). The motion continues, with the particle striking side \(OA\) at \(M_2\), \(M_3\), \( \ldots \) and striking side \(OB\) at \(N_2\), \(N_3\), \(\ldots \). Show that, if \(\theta < 60^\circ\,\), the particle reaches \(O\) in a finite time.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1484.0

Banger Comparisons: 1

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Problem source
A particle  moves on a smooth triangular horizontal surface  $AOB$ with angle 
$AOB = 30^\circ$. 
The surface  is bounded by two vertical walls 
$OA$ and $OB$ and the coefficient of restitution 
between the particle and the walls is $e$, where $e < 1$. 
The particle, which is initially at point $P$ on the surface 
and moving with velocity $u_1$, 
strikes the wall $OA$ at $M_1$, with angle $PM_1A = \theta$, and rebounds, 
with velocity $v_1$, to strike the wall $OB$ at $N_1$, 
with angle $M_1N_1B = \theta$. 
Find $e$ and $\displaystyle {v_1 \over u_1}$ in terms of $\theta$.
The motion continues, 
with the particle striking side $OA$ at $M_2$, $M_3$,   $ \ldots $ and striking 
side $OB$ at $N_2$, $N_3$,  $\ldots $. 
Show that, if $\theta < 60^\circ\,$, 
 the particle reaches $O$ in a finite time.
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