1999 Paper 2 Q10

Year: 1999
Paper: 2
Question Number: 10

Course: UFM Mechanics
Section: Momentum and Collisions 1

Difficulty: 1600.0 Banger: 1500.0

momentum collisions coefficient of restitution kinetic energy geometric series multiple collisions gravity-free space sequential impacts

Problem

$N$ particles $P_1$, $P_2$, $P_3$, $\ldots$, $P_N$ with masses $m$, $qm$, $q^2m$, $\ldots$ , ${q^{N-1}}m$, respectively, are at rest at distinct points along a straight line in gravity-free space. The particle $P_1$ is set in motion towards $P_2$ with velocity $V$ and in every subsequent impact the coefficient of restitution is $e$, where $0 < e < 1$. Show that after the first impact the velocities of $P_1$ and $P_2$ are $$ {\left({{1-eq}\over{1+q}}\right)}V \mbox{ \ \ \ and \ \ \ } {\left({{1+e}\over{1+q}}\right)}V, $$ respectively. Show that if $q \le e$, then there are exactly $N-1$ impacts and that if $q=e$, then the total loss of kinetic energy after all impacts have occurred is equal to $$ {1\over 2}{me}{\left(1-e^{N-1}\right)}{V^2}. $$

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

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Problem source

$N$ particles $P_1$, $P_2$, $P_3$, $\ldots$, $P_N$ with masses
$m$, $qm$, $q^2m$, $\ldots$ , ${q^{N-1}}m$, respectively,
are at rest 
at distinct points along a straight line in gravity-free space.
The particle $P_1$ is set in motion towards $P_2$ with velocity
$V$ and in every subsequent impact the coefficient of restitution
is $e$, where $0 < e < 1$. Show that after the first impact the
velocities of $P_1$ and $P_2$ are 
$$
{\left({{1-eq}\over{1+q}}\right)}V 
\mbox{ \ \ \ and \ \ \ } 
{\left({{1+e}\over{1+q}}\right)}V,
$$ respectively.  
Show that  if $q \le e$, then there are exactly $N-1$ impacts
and that if $q=e$, then the total loss of kinetic energy after all
impacts have occurred is equal to 
$$ 
{1\over 2}{me}{\left(1-e^{N-1}\right)}{V^2}.
$$

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