Year: 2009
Paper: 2
Question Number: 11
Course: UFM Mechanics
Section: Work, energy and Power 2
No solution available for this problem.
Of the 1000+ entries for this paper, around 920 scripts actually arrived for marking, giving another slight increase in the take-up for this paper. Of this number, five candidates scored a maximum and seventy-five achieved a scoring total of 100 or more. At the other end of the scale, almost two hundred candidates failed to reach the 40-mark mark. Otherwise, marks were spread reasonably normally across the mark range, though there were two peaks at about 45 and 65 in the distribution. It is comforting to find that the 'post-match analysis' bears out the view that I gained, quite firmly, during the marking process that there were several quantum states of mark-scoring ability amongst the candidature. Many (about one-fifth of the entry) struggled to find anything very much with which they were comfortable, and marks for these candidates were scored in 3s and 4s, with such folk often making eight or nine poor efforts at different questions without ever getting to grips with the content of any one of them. The next "ability band" saw those who either scored moderately well on a handful of questions or managed one really successful question plus a few bits-'n'-pieces in order to get up to a total in the mid-forties. To go much beyond that score required a little bit of extra talent that could lead them towards the next mark-hurdle in the mid-sixties. Thereafter, totals seemed to decline almost linearly on the distribution. Once again, it is clear that candidates need to give the questions at least a couple of minutes' worth of thought before commencing answering. Making attempts at more than the six scoring efforts permitted is a waste of valuable time, and the majority of those who do so are almost invariably the weaker brethren in the game. Many such candidates begin their efforts to individual questions promisingly, but get no more than half-a-dozen marks in before abandoning that question in favour of another – often with the replacement faring no better than its predecessor. In many such cases, the candidate's best-scoring question mark would come from their fifth, or sixth, or seventh, or …?, question attempted, and this suggests either that they do not know where their strengths lie, or that they are just not going to be of the view that they are not going to be challenged to think. And, to be fair to the setting panel, we did put some fairly obvious signposts up for those who might take the trouble to look for such things. With the pleasing number of very high totals to be found, it is clear that there are many places in which good marks were available to those with the ability to first identify them and then to persevere long enough to be able to determine what was really going on therein. It is extremely difficult to set papers in which each question is pitched at an equivalent level of difficulty. Apart from any other factors, candidates have widely differing strengths and weaknesses; one student's algebraic nuance can be the final nail in the coffin of many others, for instance. Moreover, it has seemed enormously clear to me – more particularly so since the arrival of modular A-levels – that there is absolutely no substitute for prolonged and determined practice at questions of substance. One moment's recognition of a technique at work can turn several hours of struggle into just a few seconds of polishing off, and a lack of experience is always painfully clear when marking work from candidates who are under-practised at either the art of prolonged mathematics or the science of creative problem-solving. At the other, more successful, end of the scale there were many candidates who managed to produce extraordinary amounts of outstanding work, racking up full-, or nearly full-, marks on question after question. With the marks distributed as they were, it seems that the paper was pitched appropriately at the intended level, and that it successfully managed to distinguish between the different ability-levels to be found among the candidates. 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. Moreover, many of these were clearly acts of desperation.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
A train consists of an engine and $n$ trucks.
It is travelling along a straight
horizontal section of track. The mass of the engine and of each
truck is $M$. The resistance to motion
of the engine and of each
truck is $R$, which is constant.
The maximum power at which the engine can work
is $P$.
Obtain an expression for the acceleration of the train
when its speed is $v$ and the engine is working at maximum power.
The train starts from rest with the engine working at
maximum power. Obtain an expression for the
time $T$ taken to reach a given speed $V$, and
show that this speed is only achievable if
\[
P>(n+1)RV\,.
\]
\begin{questionparts}
\item In the case when $(n+1) RV/P$ is small, use the approximation
$\ln (1-x) \approx -x -\frac12 x^2$ (valid for small $ x $)
to obtain the approximation
\[
PT\approx \tfrac 12 (n+1) MV^2\,
\]
and interpret this result.
\item In the general case, the distance moved from rest in time $T$ is $X$.
{\em Write down}, with explanation,
an equation relating $P$, $T$, $X$, $M$, $V$, $R$ and $n$ and hence
show that
\[
X= \frac{2PT - (n+1)MV^2}{2(n+1)R}
\,.
\]
\end{questionparts}
This question was the least popular question on the paper, and those trying it averaged only 6 marks on it. The most surprising aspect of it is that so few could even write a decent N2L statement to begin with, and they simply stood no chance thereafter. For those who made it to the first-order, variables-separable differential equation, the work was much more promising, though I suspect this is due to the fact that only the very able made it this far. The unpromising integration of f(v) dv, where the f(v) turned out to be a linear-over-linear algebraic fraction, was certainly unappealing to look at, but a simple substitution such as s = P – (n + 1)Rv reduces it to a very simple piece of integration. As far as I recall it, most of the inequalities in (i) were fudged, though it was very heart-warming indeed to see those excellent few who made it right to the end. It is a pity that a last minute change to the question, prior to printing, which had been intended to help candidates by giving them the final answer, then omitted the factor (n + 1) in its denominator. Fortunately, we are talking about no more than twenty of the most able (and high-scoring) candidates here; those who had explained it correctly, but then crossed-out the (n + 1) since it didn't appear on the question-paper, were given the final mark. As for those who were slightly less honest and gave the proper explanation but (presumably deliberately) didn't write the missing factor in anywhere, in order to fudge it, we were mean and didn't give them the final mark.