Year: 2002
Paper: 1
Question Number: 10
Course: UFM Mechanics
Section: Momentum and Collisions 1
No solution available for this problem.
Difficulty Rating: 1516.0
Difficulty Comparisons: 1
Banger Rating: 1470.2
Banger Comparisons: 2
A bicycle pump consists of a cylinder and a piston. The piston is pushed
in with steady speed~$u$. A particle of air moves to and fro between the
piston and the end of the cylinder, colliding perfectly elastically with the piston
and the end of the cylinder, and always moving parallel with the axis of the cylinder.
Initially, the particle is moving towards the piston at speed $v$.
Show that the speed, $v_n$, of the particle just after the
$n$th collision with the piston is given by $v_n=v+2nu$.
Let $d_n$ be the distance between the piston and the end of the cylinder
at the $n$th collision, and let $t_n$ be the time between the
$n$th and $(n+1)$th collisions. Express $d_n - d_{n+1}$ in terms
of $u$ and $t_n$, and show that
\[
d_{n+1} = \frac{v+(2n-1)u}{v+(2n+1)u} \, d_n \;.
\]
Express $d_n$ in terms of $d_1$, $u$, $v$ and $n$.
In the case $v=u$, show that $ut_n = \displaystyle \frac {d_1} {n(n+1)}$.
%%%%%Verify that $\sum\limits_1^\infty t_n = d/u$.