1995 Paper 2 Q10

Year: 1995
Paper: 2
Question Number: 10

Course: UFM Mechanics
Section: Momentum and Collisions 1

Difficulty: 1600.0 Banger: 1484.0
momentum collisions coefficient of restitution three particles sequential collisions inequality conservation of momentum change of reference frame

Problem

Three small spheres of masses \(m_{1},m_{2}\) and \(m_{3},\) move in a straight line on a smooth horizontal table. (Their order on the straight line is the order given.) The coefficient of restitution between any two spheres is \(e\). The first moves with velocity \(u\) towards the second whilst the second and third are at rest. After the first collision the second sphere hits the third after which the velocity of the second sphere is \(u.\) Find \(m_{1}\) in terms of \(m_{2},m_{3}\) and \(e\). deduce that \[ m_{2}e>m_{3}(1+e+e^{2}). \] Suppose that the relation between \(m_{1},m_{2}\) and \(m_{3}\) is that in the formula you found above, but that now the first sphere initially moves with velocity \(u\) and the other two spheres with velocity \(v\), all in the same direction along the line. If \(u>v>0\) use the first part to find the velocity of the second sphere after two collisions have taken place. (You should not need to make any substantial computations but you should state your argument clearly.)

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1484.0

Banger Comparisons: 1

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Problem source
Three small spheres of masses $m_{1},m_{2}$ and $m_{3},$ move in
a straight line on a smooth horizontal table. (Their order on the
straight line is the order given.) The coefficient of restitution
between any two spheres is $e$. The first moves with velocity $u$
towards the second whilst the second and third are at rest. After
the first collision the second sphere hits the third after which the
velocity of the second sphere is $u.$ Find $m_{1}$ in terms of $m_{2},m_{3}$
and $e$. deduce that 
\[
m_{2}e>m_{3}(1+e+e^{2}).
\]
Suppose that the relation between $m_{1},m_{2}$ and $m_{3}$ is that
in the formula you found above, but that now the first sphere initially
moves with velocity $u$ and the other two spheres with velocity $v$,
all in the same direction along the line. If $u>v>0$ use the first
part to find the velocity of the second sphere after two collisions
have taken place. (You should not need to make any substantial computations
but you should state your argument clearly.)
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