Year: 2009
Paper: 3
Question Number: 11
Course: UFM Mechanics
Section: Variable Force
No solution available for this problem.
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.1
Banger Comparisons: 2
A comet in deep space picks up mass as it travels through a
large stationary dust cloud.
It is subject to a gravitational force of magnitude
$M\!f$ acting in the direction of its motion.
When it entered the
cloud, the comet had mass $M$ and speed $V$.
After a time $t$,
it has travelled a distance $x$ through the cloud,
its mass is $M(1+bx)$, where~$b$ is a positive constant, and its speed
is $v$.
\begin{questionparts}
\item In the case when $f=0$,
write down an equation relating
$V$, $x$, $v$ and $b$.
Hence find an expression for $x$ in terms of $b$, $V$ and $t$.
\item In the case when $f$ is a non-zero constant,
use Newton's second law in the form
\[
\text{force} = \text{rate of change of momentum}
\]
to show that
\[
v = \frac{ft+V}{1+bx}\,.
\]
Hence find an expression for $x$ in terms of $b$, $V$, $f$ and $t$.
Show that it is possible, if $b$, $V$ and $f$ are suitably chosen,
for the comet to move with constant speed. Show also
that,
if the comet does not
move with constant speed, its speed tends to a constant as $t\to\infty$.
\end{questionparts}
A little less popular than question 9, the success rate was between those of the other questions in this section. Candidates tended to polarize scoring less than a third of, or very nearly all if not all of the marks. Conservation of energy, uniform acceleration and even constant speed were erroneous methods attempted. Even some good candidates carried out the integration in part (i), and then failed to solve the quadratic equation for x.