| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Topic | Projectiles |
| Type | Two moving objects interception (non-projectile) |
| Difficulty | Challenging +1.2 This is a relative velocity problem requiring vector decomposition, setting up position equations, and optimization. Part (i) involves finding closest approach using standard techniques (differentiation or perpendicular distance). Part (ii) requires solving for interception conditions with changed velocity. While multi-step and requiring careful coordinate work, these are well-practiced mechanics techniques at this level without requiring novel geometric insight. |
| Spec | 1.10g Problem solving with vectors: in geometry3.02a Kinematics language: position, displacement, velocity, acceleration3.02b Kinematic graphs: displacement-time and velocity-time |
**Question 11(i) and 11(ii)**
**(i)**
Right angled triangle with angle $\tan^{-1}\dfrac{4}{3} - 50° = 3.13°$ — B1
Shortest distance $= 5\sin 3.13° = \mathbf{0.273\ \mathrm{km}}$ — M1A1
Notes: $(20t - 5\sin 50°)^2 + (5\cos 50° - 15t)^2 \Rightarrow t = 0.1997$
**[3]**
**(ii)(a)**
A correct velocity triangle. — B1
$\dfrac{\sin\theta}{20} = \dfrac{\sin 40°}{15} \Rightarrow \theta = 58.986\ldots°$
Bearing is $\mathbf{008.99°}$ — M1A1, A1 (Accept $9°$ or $8.99°$)
**[4]**
**(ii)(b)**
$\dfrac{v}{\sin 81.013\ldots°} = \dfrac{15}{\sin 40°} \Rightarrow v = 23.049$ — M1A1
Time $= \dfrac{5}{23.049} \times 60 = \mathbf{13.0}$ minutes — M1A1 (Accept 13 minutes.)
**[4]**11 In a training exercise, a submarine is travelling due north at $15 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. The submarine commander sees his target 5 km away on a bearing of $310 ^ { \circ }$. The target is travelling due east at $20 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.\\
(i) If each of the submarine and target maintains its present course and speed, find the shortest distance between them.\\
(ii) In fact, as soon as he sees the target, the submarine commander changes course, without changing speed, so as to intercept the target as quickly as possible. Find
\begin{enumerate}[label=(\alph*)]
\item the course, in degrees, set by the submarine commander,
\item the time taken, in minutes, to intercept the target from the moment that the course changes.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2015 Q11 [11]}}