Year: 2009
Paper: 3
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A particle is projected under gravity from a point $P$ and passes
through a point $Q$. The angles of the trajectory with the positive
horizontal direction at $P$ and at $Q$ are $\theta$ and $\phi$,
respectively. The angle of elevation of $Q$ from $P$ is $\alpha$.
\begin{questionparts}
\item Show that $\tan\theta +\tan\phi = 2\tan\alpha$.
\item It is given that there is a second trajectory from $P$ to $Q$
with the same speed of projection.
The angles of this trajectory with the positive
horizontal direction at $P$ and at $Q$ are $\theta'$ and $\phi'$,
respectively.
By considering a quadratic
equation satisfied by $\tan\theta$,
show that $\tan(\theta+\theta') = -\cot\alpha$.
Show also that $\theta+\theta'=\pi+\phi+\phi'\,$.
\end{questionparts}
The most popular of the three Mechanics questions, being attempted by a sixth of the candidates, it was also the least successful, scoring only a quarter of the marks. Quite a few candidates scored nothing at all, and quite a few got the result in part (i) correctly, although by a variety of approaches, given that the uniform acceleration equations can be combined in numerous ways. However, few made any headway with the trajectory equation for part (ii).