| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Projectiles |
| Type | Two possible trajectories through point |
| Difficulty | Standard +0.8 This is a sophisticated projectiles problem requiring students to recognize that the trajectory equation becomes a quadratic in tan θ, then use discriminant analysis to find the maximum height for which two trajectories exist. This goes beyond standard projectile calculations and requires mathematical insight about the relationship between the discriminant and the envelope of trajectories, making it notably harder than typical A-level mechanics questions. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown3.02i Projectile motion: constant acceleration model |
Total: 7 marks
**Part (i):**
Trajectory formula quoted or obtained **M1**
$y = x\tan\theta - \dfrac{x^2(1 + \tan^2\theta)}{320}$ **A1** Correct including $1 + \tan^2\theta$, can be recovered (condone $g/3200$ in second term)
**Part (ii):**
Substitute **M1**
$y = 72\tan\theta - \dfrac{72^2}{320}(1 + \tan^2\theta)$ **A1**
$\alpha: 16.2t^2 - 72t + (16.2 + y) = 0$ **M1** $\beta$: Differentiate w.r.t. $\theta$
$b^2 = 4ac \Rightarrow 72^2 = 4 \times 16.2(y + 16.2)$ **M1**
$\Rightarrow y = 63.8$ **A1** Condone $y \leqslant$ [tan $\theta = 20/9$], $y = 63.8$ [exact]. Alt: Completing the square or using $-b/2a = 20/9$.
Total: **7 marks**9 A particle is projected from a point $O$ on horizontal ground with speed $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\theta$ above the horizontal.\\
(i) Write down the equation of the trajectory, in terms of $\tan \theta$.\\
(ii) The particle passes through a point whose horizontal and vertical distances from $O$ are 72 m and $y \mathrm {~m}$ respectively. By considering the equation of the trajectory as a quadratic equation in $\tan \theta$, or otherwise, find the greatest possible value of $y$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q9 [7]}}