Year: 2007
Paper: 2
Question Number: 11
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
{\sl In
this question take the acceleration due to gravity to
be
$10\,{\rm m \,s}^{-2}$ and neglect air resistance.}
The point $O$ lies in a horizontal field. The point $B$
lies $50\,$m east of $O$. A
particle is projected from $B$ at speed $25\,{\rm m\,s}^{-1}$ at an angle
$\arctan \frac12$ above the horizontal and in a direction
that makes an angle $60^\circ$
with $OB$; it passes to the north of $O$.
\begin{questionparts}
\item Taking unit vectors $\mathbf i$, $\mathbf j$ and
$\mathbf k$ in the directions east, north and vertically
upwards, respectively, find the position vector of the particle relative to
$O$ at time $t$~seconds after the particle was projected, and show that
its distance from $O$ is
\[
5(t^2- \sqrt5 t +10)\, {\rm m}.
\]
When this distance is shortest, the particle is at point $P$.
Find the position vector of $P$ and its horizontal bearing from $O$.
\item Show that the particle reaches its maximum height at $P$.
\item When the particle is at $P$, a marksman fires a bullet from $O$
directly at $P$.
The initial speed of the bullet is $350\,{\rm m\,s}^{-1}$. Ignoring the
effect of gravity on the bullet show that, when it passes
through
$P$, the distance between $P$ and the
particle is approximately~$3\,$m.
\end{questionparts}
This was almost as popular a question on Section B as Q10, being (in principle, at least) a reasonably straightforward projectiles question. Whilst many efforts were successful up to the final part, an awful lot of the attempts foundered at the very outset by failing to do the simplest of tasks: namely, noting exact values for sin θ and cos θ from tan θ = ½. It simply beggars belief that serious candidates can proceed through quite a large part of a question like this with expressions such as sin(arc tan ½) still in there! They may as well just hang out a flag which says "I'm an incompetent mathematician" on it! The three-dimensional aspect of the introduction was enough to confound most candidates attempting this question, and they were forced to resort to fiddling the given answer for the distance OP. Many attempts picked up several marks here and there throughout the question without producing anything particularly coherent, and few coped with the hazards of the last part – largely, I suspect, due to the fact that they were required to do some approximating!