Year: 2010
Paper: 3
Question Number: 11
Course: UFM Mechanics
Section: Momentum and Collisions 1
No solution available for this problem.
About 80% of candidates attempted at least five questions, and well less than 20% made genuine attempts at more than six. Those attempting more than six questions fell into three camps which were those weak candidates who made very little progress on any question, those with four or five fair solutions casting about for a sixth, and those strong candidates that either attempted 7th or even 8th questions as an "insurance policy" against a solution that seemed strong but wasn't, or else for entertainment!
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1469.5
Banger Comparisons: 2
A bullet of mass $m$ is fired horizontally with speed $u$ into a
wooden block of
mass $M$ at rest on a horizontal surface. The coefficient of friction
between the block and the surface is $\mu$. While
the bullet is moving through the block, it
experiences a constant force of resistance to its motion
of magnitude $R$, where $R>(M+m)\mu g$.
The bullet moves horizontally in the block and does not emerge from the
other side of the block.
\begin{questionparts}
\item
Show that
the magnitude, $a$, of the deceleration
of the bullet relative to the block
while the bullet is moving through the block is given by
\[
a=
\frac R m + \frac {R-(M+m)\mu g}{M}\,
.
\]
\item Show that the common speed, $v$, of the block and bullet when the
bullet stops moving through the block satisfies
\[
av = \frac{Ru-(M+m)\mu gu}M\,.
\]
\item Obtain an expression, in terms of $u$, $v$ and $a$, for the
distance moved by the block while the bullet is moving through the block.
\item Show that the total distance moved by the block is
\[
\frac{muv}{2(M+m)\mu g}\,.
\]
\end{questionparts}
Describe briefly what happens if $R< (M+m)\mu g$.
Slightly fewer attempted this than question 9, and this question was least well scored upon of any on the paper. Generally, candidates got through part (i) successfully and then either gave up or got right through the question. Common errors were the misapplication of conservation of momentum, failure to distinguish directions which led to negative signs which were then mis-handled to obtain the quoted answer in (i), and even strong candidates failing to appreciate that acceleration was constant making the later parts all susceptible to constant acceleration formulae and thus not requiring less direct approaches. The brief description at the end was usually restricted to only trivially considering the block, and few gave any thought to the bullet.