Year: 2011
Paper: 2
Question Number: 10
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
There were just under 1000 entries for paper II this year, almost exactly the same number as last year. After the relatively easy time candidates experienced on last year's paper, this year's questions had been toughened up significantly, with particular attention made to ensure that candidates had to be prepared to invest more thought at the start of each question – last year saw far too many attempts from the weaker brethren at little more than the first part of up to ten questions, when the idea is that they should devote 25-40 minutes on four to six complete questions in order to present work of a substantial nature. It was also the intention to toughen up the final "quarter" of questions, so that a complete, or nearly-complete, conclusion to any question represented a significant (and, hopefully, satisfying) mathematical achievement. Although such matters are always best assessed with the benefit of hindsight, our efforts in these areas seem to have proved entirely successful, with the vast majority of candidates concentrating their efforts on four to six questions, as planned. Moreover, marks really did have to be earned: only around 20 candidates managed to gain or exceed a score of 100, and only a third of the entry managed to hit the half-way mark of 60. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were attempted by almost all candidates; 3 and 4 by around three-quarters of them; 6, 7 and 9 by around half; the remaining questions were less popular, and some received almost no "hits". Overall, the highest scoring questions (averaging over half-marks) were 1, 2 and 9, along with 13 (very few attempts, but those who braved it scored very well). This at least is indicative that candidates are being careful in exercising some degree of thought when choosing (at least the first four) 'good' questions for themselves, although finding six successful questions then turned out to be a key discriminating factor of candidates' abilities from the examining team's perspective. Each of questions 4-8, 11 & 12 were rather poorly scored on, with average scores of only 5.5 to 6.6.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1470.2
Banger Comparisons: 2
A particle is projected from a point on a
horizontal plane, at speed
$u$ and at an angle~$\theta$ above the horizontal.
Let $H$ be
the maximum height of the particle above the plane.
Derive an expression
for $H$ in terms of $u$, $g$ and
$\theta$.
A particle $P$ is projected from a point $O$ on a smooth
horizontal plane,
at speed $u$ and at an angle~$\theta$ above the horizontal. At the
same instant, a second particle $R$ is projected horizontally from $O$
in such a way that $R$ is vertically below $P$ in the ensuing motion.
A light inextensible string of length $\frac12 H$ connects
$P$ and $R$. Show that the time that elapses
before the string becomes taut is
\[
(\sqrt2 -1)\sqrt{H/g\,}\,.
\]
When the string becomes taut, $R$ leaves the plane, the string
remaining taut. Given that $P$ and $R$ have equal masses,
determine
the total horizontal distance, $D$, travelled by $R$
from the moment its motion begins to the moment it lands on the plane
again, giving your answer
in terms of $u$, $g$ and $\theta$.
Given that $D=H$, find the value of $\tan\theta$.
This was the second most popular of the mechanics questions. The first couple of parts to the question were fairly routine in nature, but then the algebra proved too demanding in many cases, principally when it came to dealing with a quadratic equation in t which had non-numerical coefficients. Candidates also found it a struggle to know when to use g and H instead of u and θ in the working that followed. A good number of candidates understood the nature of the problem as the two particles rose and fell together, although it transpired (unexpectedly) that there was another difficult obstacle to grasp in working with two distances. Even amongst essentially fully correct solutions, very few indeed arrived at the correct final answer for tan θ.