| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2018 |
| Session | March |
| Marks | 12 |
| Topic | Projectiles |
| Type | Two projectiles meeting - 2D flight |
| Difficulty | Challenging +1.8 This is a challenging projectile motion problem requiring students to set up and solve simultaneous equations for two projectiles meeting at a point, involving double-angle formulas and algebraic manipulation across multiple steps. While the techniques are standard M1 content, the coordination of two projectiles with related angles (α and 2α) and the 7-mark allocation indicates substantial problem-solving beyond routine exercises. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| \((25\sin\alpha)t - 4.9t^2 = (15\sin(2\alpha)t - 4.9t^2\) | M1 | Use \(s = ut + \frac{1}{4}at^2\) for both, and equate |
| \(25\sin\alpha = 15\sin(2\alpha)\) | A1 | |
| \(25\sin\alpha = 30\sin\alpha\cos\alpha \Rightarrow \cos\alpha = \ldots\) | M1 | Correct use of double angle formula and attempt to solve for \(\cos\alpha\) |
| \(\cos\alpha = \frac{5}{6}\) (and \(\sin\alpha = \frac{1}{6}\sqrt{11}\)) | A1 | ; \(\alpha = 33.557\ldots°\) |
| \((25\cos\alpha)t + (15\cos(2\alpha))t = 72 \Rightarrow t = \ldots\) | M1 | Use \(s = ut\) for both, equate total to 72 and attempt to solve for \(t\) |
| \(t = 2.7\) | A1 | |
| Height of \(C\) is \((25\sin\alpha)t - 4.9t^2 = 1.59\) m | A1 | ; 1.591 028 8... |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_b = 25\cos\alpha\) | B1ft | With their value of \(\cos\alpha\); \(v_b = 20.833\ldots\) |
| \(v_v = 25\sin\alpha - 9.8t\) | B1ft | With their values of \(\sin\alpha\) and \(t\); \(v_v = \pm 12.640\ldots\) |
| \(\tan\theta = \frac{v_v}{v_b}\) | M1 | \(\theta\) is angle with horizontal; condone sign error/ambiguity for this mark |
| Direction is 31.2° below the horizontal | A1 | ; 31.247 93... |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. include the dimensions of the footballs in the model of the motion; e.g. use a more accurate value of \(g\) in the model of the motion; e.g. include air resistance in the model of the motion | B1 | DR |
## Part (i):
$(25\sin\alpha)t - 4.9t^2 = (15\sin(2\alpha)t - 4.9t^2$ | M1 | Use $s = ut + \frac{1}{4}at^2$ for both, and equate
$25\sin\alpha = 15\sin(2\alpha)$ | A1 |
$25\sin\alpha = 30\sin\alpha\cos\alpha \Rightarrow \cos\alpha = \ldots$ | M1 | Correct use of double angle formula and attempt to solve for $\cos\alpha$
$\cos\alpha = \frac{5}{6}$ (and $\sin\alpha = \frac{1}{6}\sqrt{11}$) | A1 | ; $\alpha = 33.557\ldots°$
$(25\cos\alpha)t + (15\cos(2\alpha))t = 72 \Rightarrow t = \ldots$ | M1 | Use $s = ut$ for both, equate total to 72 and attempt to solve for $t$
$t = 2.7$ | A1 |
Height of $C$ is $(25\sin\alpha)t - 4.9t^2 = 1.59$ m | A1 | ; 1.591 028 8...
## Part (ii):
$v_b = 25\cos\alpha$ | B1ft | With their value of $\cos\alpha$; $v_b = 20.833\ldots$
$v_v = 25\sin\alpha - 9.8t$ | B1ft | With their values of $\sin\alpha$ and $t$; $v_v = \pm 12.640\ldots$
$\tan\theta = \frac{v_v}{v_b}$ | M1 | $\theta$ is angle with horizontal; condone sign error/ambiguity for this mark
Direction is 31.2° below the horizontal | A1 | ; 31.247 93...
## Part (iii):
e.g. include the dimensions of the footballs in the model of the motion; e.g. use a more accurate value of $g$ in the model of the motion; e.g. include air resistance in the model of the motion | B1 | DRIn this question you must show detailed reasoning.
\includegraphics{figure_11}
A football $P$ is kicked with speed $25\,\text{m}\,\text{s}^{-1}$ at an angle of elevation $\alpha$ from a point $A$ on horizontal ground. At the same instant a second football $Q$ is kicked with speed $15\,\text{m}\,\text{s}^{-1}$ at an angle of elevation $2\alpha$ from a point $B$ on the same horizontal ground, where $AB = 72$ m. The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point $C$ (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Calculate the height of $C$ above the ground. [7]
\item Find the direction of motion of $P$ at the moment of impact. [4]
\item Suggest one improvement that could be made to the model. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2018 Q11 [12]}}