Year: 2009
Paper: 1
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
There were significantly more candidates attempting this paper again this year (over 900 in total), and the scores were pleasing: fewer than 5% of candidates failed to get at least 20 marks, and the median mark was 48. The pure questions were the most popular as usual; about two-thirds of candidates attempted each of the pure questions, with the exceptions of question 2 (attempted by about 90%) and question 5 (attempted by about one third). The mechanics questions were only marginally more popular than the probability and statistics questions this year; about one quarter of the candidates attempted each of the mechanics questions, while the statistics questions were attempted by about one fifth of the candidates. A significant number of candidates ignored the advice on the front cover and attempted more than six questions. In general, those candidates who submitted answers to eight or more questions did fairly poorly; very few people who tackled nine or more questions gained more than 60 marks overall (as only the best six questions are taken for the final mark). This suggests that a skill lacking in many students attempting STEP is the ability to pick questions effectively. This is not required for A-levels, so must become an important part of STEP preparation. Another "rubric"-type error was failing to follow the instructions in the question. In particular, when a question says "Hence", the candidate must make (significant) use of the preceding result(s) in their answer if they wish to gain any credit. In some questions (such as question 2), many candidates gained no marks for the final part (which was worth 10 marks) as they simply quoted an answer without using any of their earlier work. There were a number of common errors which appeared across the whole paper. These included a noticeable weakness in algebraic manipulations, sometimes indicating a serious lack of understanding of the mathematics involved. As examples, one candidate tried to use the misremembered identity cos β = sin √(1 − β²), while numerous candidates made deductions of the form "if a² + b² = c², then a + b = c" at some point in their work. Fraction manipulations are also notorious in the school classroom; the effects of this weakness were felt here, too. Another common problem was a lack of direction; writing a whole page of algebraic manipulations with no sense of purpose was unlikely to either reach the requested answer or gain the candidate any marks. It is a good idea when faced with a STEP question to ask oneself, "What is the point of this (part of the) question?" or "Why has this (part of the) question been asked?" Thinking about this can be a helpful guide. One aspect of this is evidenced by pages of formulæ and equations with no explanation. It is very good practice to explain why you are doing the calculation you are, and to write sentences in English to achieve this. It also forces one to focus on the purpose of the calculations, and may help avoid some dead ends. Finally, there is a tendency among some candidates when short of time to write what they would do at this point, rather than using the limited time to actually try doing it. Such comments gain no credit; marks are only awarded for making progress in a question. STEP questions do require a greater facility with mathematics and algebraic manipulation than the A-level examinations, as well as a depth of understanding which goes beyond that expected in a typical sixth-form classroom.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
Two particles $P$ and $Q$ are projected simultaneously from points
$O$ and $D$, respectively, where~$D$ is a distance $d$ directly
above $O$. The initial speed of $P$ is $V$ and its angle of projection
{\em above} the horizontal is $\alpha$. The initial speed of $Q$ is
$kV$, where $k>1$, and its angle of projection {\em below} the horizontal
is $\beta$. The particles collide at time $T$ after projection.
Show that $\cos\alpha = k\cos\beta$ and that
$T$ satisfies the equation
\[
(k^2-1)V^2T^2 +2dVT\sin\alpha -d^2 =0\,.
\]
Given that the particles collide when $P$ reaches its maximum height,
find an expression for~$\sin^2\alpha$ in terms of $g$, $d$, $k$ and
$V$,
and deduce that
\[
gd\le (1+k)V^2\,.
\]
The start of this question was found to be fairly approachable. Most candidates were able to draw a correct sketch of the situation. However, a significant number became unstuck at this very early stage, getting the directions incorrect (for example, having Q at an angle of β above the horizontal) or swapping P and Q or worse. The next challenge was to correctly apply the "suvat" equations (motion with constant acceleration) to this situation. This was done successfully for P by most candidates, although some made sign errors; it is vital to always indicate which direction is positive in problems such as these. The motion of Q, however, caused many problems: what is s when the motion does not start from the ground? Some candidates determined the displacement from the starting position and then tried adding or subtracting d, others introduced d later on, while others seemed to ignore d entirely. The general principle is to be clear about the meaning of one's symbols; being explicit whether s is the vertical displacement upwards or downwards from the ground or from D would have averted many of the problems. Most candidates were able to show that cos α = k cos β, but the quadratic proved too much for the majority of them. There were many attempts, but because of the earlier errors in calculating the displacements of the particles or mistakes in algebraic manipulations, the quadratic did not appear. Many did not realise that it was necessary to use the earlier result about the cosines, while others asserted that sin α = k sin β. About one-fifth of the attempts at the question reached the final part of the question. Of those, most were able to determine a formula for T, but many became stuck at this point. Once again, there were frequent attempts to rearrange formulæ, but few were successful. Only a handful of candidates managed to progress beyond the requirement that sin² α ⩽ 1 to reach the desired conclusion.