Year: 2018
Paper: 1
Question Number: 9
Course: UFM Mechanics
Section: Work, energy and Power 1
In order to get the fullest picture, this document should be read in conjunction with the question paper, the marking scheme and (for comments on the underlying purpose and motivation for finding the right solution-approaches to questions) the Hints and Solutions document. The purpose of the STEPs is to learn what students are able to achieve mathematically when applying the knowledge, skills and techniques that they have learned within their standard A-level (or equivalent) courses … but seldom within the usual range of familiar settings. STEP questions require candidates to work at an extended piece of mathematics, often with the minimum of specific guidance, and to make the necessary connections. This requires a very different mind-set to that which is sufficient for success at A-level, and the requisite skills tend only to develop with prolonged and determined practice at such longer questions. One of the most crucial features of the STEPs is that the routine technical and manipulative skills are almost taken for granted; it is necessary for candidates to produce them with both speed and accuracy so that the maximum amount of time can be spent in thinking their way through the problem and the various hurdles and obstacles that have been set before them. Most STEP questions begin by asking the solver to do something relatively routine or familiar before letting them loose on the real problem. Almost always, such an opening has not been put there to allow one to pick up a few easy marks, but rather to point the solver in the right direction for what follows. Very often, the opening result or technique will need to be used, adapted or extended in the later parts of the question, with the demands increasing the further on that one goes. So a candidate should never think that they are simply required to 'go through the motions'; rather they will, sooner or later, be required to show either genuine skill or real insight in order to make a reasonably complete effort. The more successful candidates are the ones who manage to figure out how to move on from the given starting-point. Finally, reading through a finished solution is often misleading – even unhelpful – unless you have attempted the problem for yourself. This is because the thinking has been done for you. So, when you read through the report and look at the solutions (either in the mark scheme or the Hints and Solutions booklet), try to figure out how you could have arrived at the solution, learn from your mistakes and pick up as many tips as you can whilst working through past paper questions. This year far too many candidates wasted time by attempting more than six questions, with many of these candidates picking up 0-4 marks on several 'false starts' which petered out the moment some understanding was required. There were almost 2000 candidates for this SI paper. Almost one-sixth of candidates failed to reach a total of 30 and around two-thirds fell below half-marks overall. This highlights the fact that many candidates don't find this test an easy one. At the other end of the spectrum, almost one-in-ten managed a total of 84 out of 120 – these candidates usually marked out by their ability to complete whole questions – with almost 4% of the entry achieving the highly praiseworthy feat of getting into three-figures with their overall score. The paper is constructed so that question 1 is very approachable indeed, the intention being to get everyone started with some measure of success; unsurprisingly, Q1 was the most popular question of all, with almost all candidates attempting it, and it also turned out to be the most successful question on the paper with a mean score of more than 15 out of 20. Around 7% of candidates didn't make any kind of attempt at it at all. In order of popularity, Q1 was followed by Qs. 2, 7, 4 and 3. Indeed, it was the pure maths questions in Section A that attracted the majority of attention from candidates, with the most popular applied question (Q9, mechanics) still getting fewer 'hits' than the least popular pure question (Q5). Questions 10, 11 and 13 proved to attract very little attention from candidates and many of the attempts were minimal.
Difficulty Rating: 1516.0
Difficulty Comparisons: 1
Banger Rating: 1500.0
Banger Comparisons: 0
A straight road leading to my house consists of two sections. The first section is inclined downwards at a constant angle $\alpha$ to the horizontal and ends in traffic lights; the second section is inclined upwards at an angle $\beta$ to the horizontal and ends at my house. The distance between the traffic lights and my house is $d$.
I have a go-kart which I start from rest, pointing downhill, a distance $x$ from the traffic lights on the downward-sloping section. The go-kart is not powered in any way, all resistance forces are negligible, and there is no sudden change of speed as I pass the traffic lights. Given that I reach my house, show that $x \sin \alpha\ge d \sin\beta\,$.
Let $T$ be the total time taken to reach my house.
Show that
\[
\left(\frac{g\sin\alpha}2 \right)^{\!\frac12} T =
(1+k) \sqrt{x} - \sqrt{k^2 x -kd\;}
\,,
\]
where $k = \dfrac{\sin\alpha}{\sin\beta}\,$.
Hence determine, in terms of $d$ and $k$, the value of $x$ which minimises $T$.
[You need not justify the fact that the stationary value is a minimum.]Applying conservation of energy, since there are no external forces (other than gravity) the condition to reach the house (with any speed) is the initial GPE is larger than the final GPE, ie:
\begin{align*}
&& m g x \sin \alpha &\geq m g d \sin \beta \\
\Rightarrow && x \sin \alpha &\geq d \sin \beta
\end{align*}
Let $T_1$ be the time taken on the downward section, and $T_2$ the time taken on the upward section, then:
\begin{align*}
&& s &= ut + \frac12 a t^2 \\
\Rightarrow && x &= \frac12 g \sin \alpha T_1^2 \\
\Rightarrow && T_1^2 &= \frac{2x}{g \sin \alpha} \\
&& v &= u + at \\
\Rightarrow && v &= T_1 g \sin \alpha \\
&& mg x \sin \alpha &= mg d \sin \beta + \frac12 m w^2 \\
\Rightarrow && w &= \sqrt{2(x \sin \alpha - d \sin \beta)} \\
&& w &= v - g \sin \beta T_2 \\
\Rightarrow && T_2 &= \frac{v - w}{g \sin \beta} \\
\Rightarrow && T &= T_1 + T_2 \\
&&&= \sqrt{\frac{2x}{g \sin \alpha}} + \frac{\sqrt{\frac{2x}{g \sin \alpha}} g \sin \alpha- \sqrt{2(x \sin \alpha - d \sin \beta)}}{g \sin \beta} \\
&&&= \left ( \frac{2}{g \sin \alpha} \right)^{\tfrac12} \left ( \sqrt{x} + \sqrt{x}k - \sqrt{k^2x-kd}\right)
\end{align*}
Differentiating wrt to $x$, we obtain:
\begin{align*}
&& \frac{\d T}{\d x} &= C(-(1+k)x^{-1/2}+k^2(k^2 x - kd)^{-1/2}) \\
\text{set to }0: && 0 &= k^2(k^2 x - kd)^{-1/2} - (1+k)x^{-1/2} \\
\Rightarrow && \sqrt{x} k^2 &= \sqrt{k^2x - kd} (1+k) \\
\Rightarrow && x k^4 &= (k^2x-kd)(1+k)^2 \\
\Rightarrow && x(k^4-k^2(1+k)^2) &= -kd(1+k)^2 \\
\Rightarrow && x(2k^2+k) &= d \\
\Rightarrow && x &= \frac{d}{(2k^2+k)}
\end{align*}
This was the most popular of the applied questions, drawing more than 700 attempts. It was, after Qs.1 & 2, the highest-scoring question, one of only the three to average more than 10 out of 20. Most candidates realised that an energy argument was useful in the first part of the question, but there were some spurious reasons given for the direction of the inequality. Candidates were then generally able to apply kinematics formulae to make progress; however, reaching the required form proved challenging. In particular, candidates were often unclear about the sign conventions they were using. When taking square roots in mechanics – either directly or in the context of the quadratic equation – the plus or minus solutions often have two physical interpretations and candidates were very bad at explaining why they were selecting one of the signs. The algebra required to get to the given form was beyond most candidates. However, they could still have used the given form to answer the last part using some basic calculus.