Year: 2007
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
Although the paper was by no means an easy one, it was generally found a more accessible paper than last year's, with most questions clearly offering candidates an attackable starting-point. The candidature represented the usual range of mathematical talents, with a pleasingly high number of truly outstanding students; many more who were able to demonstrate a thorough grasp of the material in at least three questions; and the few whose three-hour long experience was unlikely to have been a particularly pleasant one. However, even for these candidates, many were able to make some progress on at least two of the questions chosen. Really able candidates generally produced solid attempts at five or six questions, and quite a few produced outstanding efforts at up to eight questions. In general, it would be best if centres persuaded candidates not to spend valuable time needlessly in this way – it is a practice that is not to be encouraged, as it uses valuable examination time to little or no avail. Weaker brethren were often to be found scratching around at bits and pieces of several questions, with little of substance being produced on more than a couple. It is an important examination skill – now more so than ever, with most candidates now not having to employ such a skill on the modular papers which constitute the bulk of their examination experience – for candidates to spend a few minutes at some stage of the examination deciding upon their optimal selection of questions to attempt. As a rule, question 1 is intended to be accessible to all takers, with question 2 usually similarly constructed. In the event, at least one – and usually both – of these two questions were among candidates' chosen questions. These, along with questions 3 and 6, were by far the most popularly chosen questions to attempt. The majority of candidates only attempted questions in Section A (Pure Maths), and there were relatively few attempts at the Applied Maths questions in Sections B & C, with Mechanics proving the more popular of the two options. It struck me that, generally, the working produced on the scripts this year was rather better set-out, with a greater logical coherence to it, and this certainly helps the markers identify what each candidate thinks they are doing. Sadly, this general remark doesn't apply to the working produced on the Mechanics questions, such as they were. As last year, the presentation was usually appalling, with poorly labelled diagrams, often with forces missing from them altogether, and little or no attempt to state the principles that the candidates were attempting to apply.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A solid right circular cone, of mass $M$, has
semi-vertical angle $\alpha$ and smooth surfaces.
It stands with its base on
a smooth horizontal table.
A particle of mass $m$
is projected
so that it strikes the curved surface of the cone
at speed $u$.
The coefficient of restitution
between the particle and the cone is $e$.
The impact has no rotational effect on the cone
and the cone has no vertical velocity
after the impact.
\begin{questionparts}
\item The particle strikes the cone in the direction of the normal at
the point of impact. Explain why the trajectory of the particle
immediately
after the impact is parallel to the normal to the surface of the cone.
Find an expression, in terms of $M$, $m$,
$\alpha$, $e$ and $u$, for the speed at which the cone slides
along the table immediately after impact.
\item
If instead the particle falls vertically onto the cone, show that the speed $w$
at which the cone slides
along the table immediately after impact is given by
\[
w= \frac{mu(1+e)\sin\alpha\cos\alpha}{M+m\cos^2\alpha}\,.
\]
Show also that the value of $\alpha$ for which $w$ is greatest is given by
\[
\cos \alpha = \sqrt{ \frac{M}{2M+m}}\ .
\]
\end{questionparts}
This was the least popular of the Mechanics questions, perhaps because it commenced with a request for an explanation. As mentioned already, a clearly labelled diagram or two would have been enormously helpful here! The fact that there are only the two mechanical principles being employed here should have made it an easy question, but efforts were generally very poor.