Year: 2013
Paper: 2
Question Number: 10
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
All questions were attempted by a significant number of candidates, with questions 1 to 3 and 7 the most popular. The Pure questions were more popular than both the Mechanics and the Probability and Statistics questions, with only question 8 receiving a particularly low number of attempts within the Pure questions and only question 11 receiving a particularly high number of attempts.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A particle is projected
at an angle of elevation $\alpha$ (where $\alpha>0$) from a point
$A$ on horizontal ground.
At a general
point in its trajectory the angle of elevation of the particle
from $A$ is $\theta$ and
its direction of motion is at an angle $\phi$ above the horizontal
(with $\phi\ge0$ for the first half of the trajectory and $\phi\le0$
for the second half).
Let $B$ denote the point on the trajectory at which $\theta = \frac12 \alpha$
and let $C$ denote the point on the trajectory at which
$\phi = -\frac12\alpha$.
\begin{questionparts}
\item Show that, at a general point on the trajectory,
$2\tan\theta = \tan \alpha + \tan\phi\,$.
\item Show that, if $B$ and $C$ are the same point, then
$ \alpha = 60^\circ\,$.
\item Given that
$\alpha < 60^\circ\,$,
determine whether the particle reaches the point $B$ first or the
point $C$ first.
\end{questionparts}
This was the least popular of the Mechanics questions. The first part of the question was generally well answered and many candidates were able to apply the result of part (i) to the particular case identified in part (ii). Part (iii) was found to be more challenging, but some candidates did manage to provide a convincing argument for their answer.