Year: 2017
Paper: 2
Question Number: 10
Course: UFM Mechanics
Section: Variable Force
This year's paper was, perhaps, slightly more straightforward than usual, with more helpful guidance offered in some of the questions. Thus the mark required for a "1", a Distinction, was 80 (out of 120), around ten marks higher than that which would customarily be required to be awarded this grade. Nonetheless, a three‐figure mark is still a considerable achievement and, of the 1330 candidates sitting the paper, there were 89 who achieved this. At the other end of the scale, there were over 350 who scored 40 or below, including almost 150 who failed to exceed a total score of 25. As a general strategy for success in a STEP examination, candidates should be looking to find four "good" questions to work at (which may be chosen freely by the candidates from a total of 13 questions overall). It is unfortunately the case that so many low‐scoring candidates flit from one question to another, barely starting each one before moving on. There needs to be a willingness to persevere with a question until a measure of understanding as to the nature of the question's purpose and direction begins to emerge. Many low‐scoring candidates fail to deal with those parts of questions which cover routine mathematical processes ‐ processes that should be standard for an A‐level candidate. The significance of the "rule of four" is that four high‐scoring questions (15‐20 marks apiece) obtains you up to around the total of 70 that is usually required for a "1"; and with a couple of supporting starts to questions, such a total should not be beyond a good candidate who has prepared adequately. This year, significantly more than 10% of candidates failed to score at least half marks on any one question; and, given that Q1 (and often Q2 also) is (are) specifically set to give all candidates the opportunity to secure some marks, this indicates that these candidates are giving up too easily. Mathematics is about more than just getting to correct answers. It is about communicating clearly and precisely. Particularly with "show that" questions, candidates need to distinguish themselves from those who are just tracking back from given results. They should also be aware that convincing themselves is not sufficient, and if they are using a result from 3 pages earlier, they should make this clear in their working. A few specifics: In answers to mechanics questions, clarity of diagrams would have helped many students. If new variables or functions are introduced, it is important that students clearly define them. One area which is very important in STEP but which was very poorly done is dealing with inequalities. Although a wide range of approaches such as perturbation theory were attempted, at STEP level having a good understanding of the basics – such as changing the inequality if multiplying by a negative number – is more than enough. In fact, candidates who used more advanced methods rarely succeeded.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A car of mass $m$ makes a journey of distance $2d$
in a straight line.
It experiences air resistance and rolling resistance
so that the total resistance to motion when it is moving with
speed $v$ is $Av^2 +R$, where $A$ and $R$ are constants.
The car starts from rest and
moves with constant acceleration $a$ for a distance $d$.
Show that the work done by the engine
for this half of the journey
is
\[
\int_0^d (ma+R+Av^2) \, \d x
\]
and that it can be written
in the form
\[
\int_0^w \frac {(ma+R+Av^2)v}a\; \d v
\,,
\]
where $w =\sqrt {2ad\,}\,$.
For the second half of the journey, the acceleration of
the car is $-a$.
\begin{questionparts}
\item
In the case $R>ma$,
show that the work done by the
engine for the whole journey is
\[
2Aad^2 + 2Rd
\,.
\]
\item
In the case $ma-2Aad< R< ma$, show that at a certain speed
the driving
force required to maintain the constant acceleration
falls to zero.
Thereafter, the engine does no work
(and the driver applies the brakes to maintain
the constant acceleration).
Show that the work done by the engine for the whole journey is
\[
2Aad^2 + 2 Rd
+ \frac{(ma-R)^2}{4Aa}
\,
.\]
\end{questionparts}The force delivered by the engine must be $ma + R + Av^2$, (so the net force is $ma$). Therefore the work done is $\displaystyle \int_0^d F \d x = \int_0^d (ma + R + Av^2) \d x$
Notice that $a = v \frac{\d v}{\d x} \Rightarrow \frac{a}{v} = \frac{\d v}{\d x}$ and so
\begin{align*}
&& WD &= \int_0^d (ma + R + Av^2) \d x \\
&&&= \int_{x=0}^{x=d} (ma + R + Av^2) \frac{v}{a} \frac{\d v}{\d x} \d x \\
&&&= \int_{x=0}^{x=d} \frac{ (ma + R + Av^2)v}{a} \d v \\
\end{align*}
Also notice that if we move with constant acceleration from rest for a distance $d$ the final speed is $v^2 = 2ad \Rightarrow v = \sqrt{2ad}$
\begin{questionparts}
\item For the second part of the journey, the engine will be putting out a force of $-ma+R+Av^2>0$, and the car will have a final speed of $0$
\begin{align*}
WD &= \int_0^{w} \frac{(ma+R+Av^2)v}{a} \d v + \int_w^0 \frac{(-ma+R+Av^2)v}{-a} \d v \\
&= \int_0^w \frac{2(Rv+Av^3)}{a} \d v \\
&= \frac{Rw^2+\frac12Aw^4}{a} \\
&= \frac{R2ad+\frac12A4a^2d^2}{a} \\
&= 2Rd + 2Aad^2
\end{align*}
\item If $ma - 2Aad < R < ma$ then the driving force is still $-ma+R+Av^2$ which is positive when $v = \sqrt{2as}$ but negative when $v = 0$, and therefore at some point in-between the driving force must be $0$.
The engine will stop working when $-ma+R+Av^2 =0 \Rightarrow v = \sqrt{\frac{ma-R}{A}}$ so
\begin{align*}
WD &= \int_0^w \frac{(ma+R+Av^2)v}{a} \d v + \int_w^{ \sqrt{\frac{ma-R}{A}}} \frac{(-ma+R+Av^2)v}{-a} \d v \\
&= \int_0^w \frac{2(R+Av^2)v}{a} \d v - \int_0^{\sqrt{\frac{ma-R}{A}}} \frac{(-ma+R+Av^2)v}{a}\d v \\
&= 2Aad^2+2Rd + \frac1a\left (\frac12(R-ma)\frac{ma-R}{A} + \frac{A}{4}\left ( \frac{ma-R}{A}\right)^2 \right) \\
&= 2Aad^2+2Rd - \frac{(ma-R)^2}{Aa}\left (-\frac12+ \frac14 \right) \\
&= 2Aad^2+2Rd - \frac{(ma-R)^2}{4Aa}
\end{align*}
\end{questionparts}
This was another very unpopular mechanics question. Many candidates who might have thought they made considerable progress did not score very well because they failed to communicate clearly the details required to "show" the given results.