Year: 2024
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
Many candidates produced good solutions to the questions, with the majority of candidates opting to focus on the pure questions of the paper. Candidates demonstrated very good ability, particularly in the area of manipulating algebra. Many candidates produced clear diagrams which in many cases meant that they were more successful in their attempts at their questions than those who did not do so. The paper also contained a number of places where the answer to be reached was given in the question. In such cases, candidates must be careful to ensure that they provide sufficient evidence of the method used to reach the result in order to gain full credit.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A long straight trench, with rectangular cross section, has been dug in otherwise horizontal ground. The width of the trench is $d$ and its depth $2d$. A particle is projected at speed $v$, where $v^2 = \lambda dg$, at an angle $\alpha$ to the horizontal, from a point on the ground a distance $d$ from the nearer edge of the trench. The vertical plane in which it moves is perpendicular to the trench.
\begin{questionparts}
\item The particle lands on the base of the trench without first touching either of its sides.
\begin{enumerate}
\item By considering the vertical displacement of the particle when its horizontal displacement is $d$, show that $(\tan\alpha - \lambda)^2 < \lambda^2 - 1$ and deduce that $\lambda > 1$.
\item Show also that $(2\tan\alpha - \lambda)^2 > \lambda^2 + 4(\lambda - 1)$ and deduce that $\alpha > 45^\circ$.
\end{enumerate}
\item Show that, provided $\lambda > 1$, $\alpha$ can always be chosen so that the particle lands on the base of the trench without first touching either of its sides.
\end{questionparts}
About one-third of the candidates attempted this question. In some cases the diagrams that were drawn showed that candidates had not taken care to fully understand the description of the situation. Part (i)(a) was completed well by many candidates, but some did not recognise the need to use a trigonometric identity to obtain a form which is a function of tan α which is a common technique in questions of this form. Part (i)(b) was mostly done well, although most did not score fully on the final deduction, often not applying the condition that λ > 1 correctly and reaching two cases, one of which they could not rule out. Very few candidates produced a good solution to part (ii), but those who managed to identify the correct starting point generally produced good solutions.