Year: 2020
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
There were just over 800 entries for this paper, and good solutions were seen to all of the questions. Candidates should be aware of the need to provide clear explanations of their reasoning throughout the paper (and particularly in questions where the result to be shown is given in the question). Short explanatory comments at key points in solutions can be very helpful in this regard, as can clearly drawn diagrams of the situation described in the question. The paper included a few questions where a statement of the form "A if and only if B" needed to be proven – candidates should be aware of the meaning of such statements and make sure that both directions of the implication are covered clearly. In general, candidates who performed better on the questions in this paper recognised the relationships between the different parts of each question and were able to adapt methods used in earlier parts when working on the later sections of the question.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Point $A$ is a distance $h$ above ground level and point $N$ is directly below $A$ at ground level. Point $B$ is also at ground level, a distance $d$ horizontally from $N$. The angle of elevation of $A$ from $B$ is $\beta$. A particle is projected horizontally from $A$, with initial speed $V$. A second particle is projected from $B$ with speed $U$ at an acute angle $\theta$ above the horizontal. The horizontal components of the velocities of the two particles are in opposite directions. The two particles are projected simultaneously, in the vertical plane through $A$, $N$ and $B$.
Given that the two particles collide, show that
\[d\sin\theta - h\cos\theta = \frac{Vh}{U}\]
and also that
\begin{questionparts}
\item $\theta > \beta$;
\item $U\sin\theta \geqslant \sqrt{\dfrac{gh}{2}}$;
\item $\dfrac{U}{V} > \sin\beta$.
\end{questionparts}
Show that the particles collide at a height greater than $\frac{1}{2}h$ if and only if the particle projected from $B$ is moving upwards at the time of collision.
This was a question that was found to be difficult. In general, this question was not attempted well, with very few candidates progressing past the first section. Most candidates managed to pick up all the marks in the initial section of the question. However, a significant minority of students could not set up the problem correctly or knew a lot of linear acceleration (suvat) equations but could not apply them correctly (for example mistaking displacement for position) and received zero marks. Some candidates eliminated t in favour of x and could not progress to the last calculation. Around half the candidates picked up full marks for part (i). However, many candidates tried to reason with words – almost always unsuccessfully, often believing that the particle projected from point A could not pass through the line AB. Most of the candidates received zero marks for part (ii), failing to realise that the result follows from the height of the particle at the time of collision being non-negative. Some tried to use conservation of momentum or energy, or the equation v² = u² + 2as due to the answer being suggestive of velocity squared. Candidates who were able to progress well on this part generally achieved all of the marks. Very few candidates progressed to part (iii) and the attempts were often poor. Candidates who did know how to proceed to the result often did not justify bounds they used to obtain inequalities. A significant number of candidates attempted the final part of the question having omitted earlier parts. In many cases candidates did not fully appreciate the requirements when asked to show a statement of the form "A if and only if B".