Year: 2014
Paper: 2
Question Number: 10
Course: LFM Pure and Mechanics
Section: Projectiles
No solution available for this problem.
There were good solutions presented to all of the questions, although there was generally less success in those questions that required explanations of results or the use of diagrams and graphs to reach the solution. Algebraic manipulation was generally well done by many of the candidates although a range of common errors such as confusing differentiation and integration and simple arithmetic slips were evident. Candidates should also be advised to use the methods that are asked for in questions unless it is clear that other methods will be accepted (such as by the use of the phrase "or otherwise").
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A particle is projected from a point $O$ on horizontal ground
with initial speed $u$ and at an angle of
$\theta$ above the ground. The motion takes place in the
$x$-$y$ plane, where the $x$-axis is horizontal, the $y$-axis is
vertical and
the origin is $O$.
Obtain the Cartesian equation of the particle's trajectory in
terms of $u$, $g$ and~$\lambda$, where $\lambda=\tan\theta$.
Now consider the trajectories for different values of $\theta$
with $u$~fixed. Show that for a given value of~$x$, the
coordinate~$y$ can take all values up to a maximum value,~$Y$,
which you should determine as a function of $x$, $u$ and~$g$.
Sketch a graph of $Y$ against $x$ and indicate on your graph
the set of points that can be reached by a particle projected
from $O$ with speed $u$.
Hence find the furthest distance from $O$ that can be achieved
by such a projectile.
This was the most popular of the mechanics questions and also the one that had the best average score, although candidates did struggle to get very high marks on the question particularly on the final parts. The first part of the question asks for a derivation of the equation for the trajectory which was familiar to many candidates, although in some cases the result was obtained by stating that it is a parabola and knowledge of the maximum value and the range. Many candidates who successfully obtained the Cartesian equation then struggled with the differentiation with respect to θ, instead finding the maximum height for a constant value of θ. Unfortunately, this made the remainder of the question insoluble. Some candidates decided to differentiate with respect to x instead, which did not cause any serious problems, although it did require more work. A few candidates used the discriminant rather than differentiation, but did not provide any justification of this method. Candidates were able to draw the graph, but many did not label the area that was asked for in the question. Those who reached the final part of the question and considered the distance function for the position during the flight used differentiation to work out the greatest distance. However, many did not realise that the maximum value of a function can be achieved at an end-point of the domain even with a derivative that is non-zero.