2014 Paper 1 Q9
Year: 2014
Paper: 1
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
Problem
Solution
Examiner's report
— 2014 STEP 1, Question 9More than 1800 candidates sat this paper, which represents another increase in uptake for this STEP paper. The impression given, however, is that many of these extra candidates are just not sufficiently well prepared for questions which are not structured in the same way as are the A-level questions that they are, perhaps, more accustomed to seeing. Although STEP questions try to give all able candidates "a bit of an intro." into each question, they are not intended to be easy, and (at some point) imagination and real flair (as well as determination) are required if one is to score well on them. In general, it is simply not possible to get very far into a question without making some attempt to think about what is actually going on in the situation presented therein; and those students who expect to be told exactly what to do at each stage of a process are in for a shock. Too many candidates only attempt the first parts of many questions, restricting themselves to 3-6 marks on each, rather than trying to get to grips with substantial portions of work – the readiness to give up and try to find something else that is "easy pickings" seldom allows such candidates to acquire more than 40 marks (as was the case with almost half of this year's candidature, in fact). Poor preparation was strongly in evidence – curve-sketching skills were weak, inequalities very poorly handled, algebraic capabilities (especially in non-standard settings) were often pretty poor, and the ability to get to grips with extended bits of working lacking in the extreme; also, an unwillingness to be imaginative and creative, allied with a lack of thoroughness and attention to detail, made this a disappointing (and, possibly, very uncomfortable) experience for many of those students who took the paper. On the other side of the coin, there was a very pleasing number of candidates who produced exceptional pieces of work on 5 or 6 questions, and thus scored very highly indeed on the paper overall. Around 100 of them scored 90+ marks of the 120 available, and they should be very proud of their performance – it is a significant and noteworthy achievement.
Rating Information
Difficulty Rating: 1516.0
Difficulty Comparisons: 1
Banger Rating: 1500.0
Banger Comparisons: 0
Show LaTeX source
Problem source
A particle of mass $m$ is projected due east at speed $U$ from a point on horizontal ground at an angle $\theta$ above the horizontal, where $0 < \theta < 90^\circ$. In addition to the gravitational force $mg$, it experiences a horizontal force of magnitude $mkg$, where $k$ is a positive constant, acting due west in the plane of motion of the particle. Determine expressions in terms of $U$, $\theta$ and $g$ for the time, $T_H$, at which the particle reaches its greatest height and the time, $T_L $, at which it lands.
Let $T = U\cos\theta /(kg)$. By considering the relative magnitudes of $T_H$, $T_L $ and $T$, or otherwise, sketch the trajectory of the particle in the cases $k\tan\theta<\frac12$, $\frac12 < k\tan\theta<1$, and $k\tan\theta>1$.
What happens when $k\tan\theta =1$?Solution source
\begin{align*}
&& v_{\uparrow} &= U\sin \theta - g t \\
\Rightarrow && T_H &= \frac{U \sin \theta}{g} \\
\\
&& s_{\uparrow} &= U \sin \theta t - \frac12 g t^2 \\
\Rightarrow && 0 &= U\sin \theta T_L - \frac12 g T_L^2 \\
&& T_L &= \frac{2 U \sin \theta}{g}
\end{align*}
$T = U\cos \theta / (kg)$ is the point when the particle's horizontal motion is reversed.
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){1/((#1)*(#1))};
\def\xl{-5};
\def\xu{11.5};
\def\yl{-0.01};
\def\yu{2};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{10/\xrange}
\pgfmathsetmacro{\yscale}{10/\yrange}
% Define the reusable styles to keep code clean
\tikzset{
x=\xscale cm, y=\yscale cm,
axis/.style={thick, draw=black!80, -{Stealth[scale=1.2]}},
grid/.style={thin, dashed, gray!30},
curveA/.style={very thick, color=cyan!70!black, smooth},
curveB/.style={very thick, color=orange!90!black, smooth},
curveC/.style={very thick, color=red!90!black, smooth},
dot/.style={circle, fill=black, inner sep=1.2pt},
labelbox/.style={fill=white, inner sep=2pt, rounded corners=2pt} % Protects text from lines
}
% Draw background grid
% \draw[grid] (\xl,\yl) grid[step=1] (\xu,\yu);
% Set up axes
\draw[axis] (\xl,0) -- (\xu,0) node[right, black] {$x$};
% \draw[axis] (0,\yl) -- (0,\yu) node[above, black] {$y$};
% Define the bounding region with clip
\begin{scope}
\clip (\xl,\yl) rectangle (\xu,\yu);
\draw[curveA, domain=0:{10*2*sin(30)/10}, samples=151]
plot ({10*cos(30)*\x - 2 * \x * \x}, {10*sin(30)*\x - 5 * \x * \x});
\draw[curveB, domain=0:{10*2*sin(30)/10}, samples=151]
plot ({10*cos(30)*\x - 5.1 * \x * \x}, {10*sin(30)*\x - 5 * \x * \x});
\draw[curveC, domain=0:{10*2*sin(30)/10}, samples=151]
plot ({10*cos(30)*\x - 12 * \x * \x}, {10*sin(30)*\x - 5 * \x * \x});
\end{scope}
\end{tikzpicture}
\end{center}
When $k\tan \theta = 1$ it lands exactly where it started.
The mechanics questions in section B always seem to prove more popular than those in the probability & statistics section C, and this was again the case this year. Q9 drew around 700 attempts. The key point in Q9 was to realise that the given expression for T meant that T was the time when the particle "turned round" horizontally. Those candidates who spotted this then had the opportunity to score lots of marks, as the sketches related to the point at which the particle turned back relative to its highest point and its landing-point. Those who approached the problem from a purely algebraic direction usually struggled with the significances of the three (four) cases. Q9's mean score was just under 6½/20.