| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projectile clearing obstacle |
| Difficulty | Standard +0.8 This is a substantial projectile motion problem requiring trajectory equation derivation, optimization to find minimum velocity, verification of maximum height, and range calculation. While the techniques are standard M2 content (projectiles at 45°), the multi-part nature, the 7-mark optimization in part (a), and the need to connect multiple concepts (trajectory, maximum height, range) make this more demanding than typical textbook exercises. It requires careful algebraic manipulation and physical insight about when the minimum occurs. |
| Spec | 3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(x = (u\cos 45°)t\), \(y = (u\sin 45°)t - 4.9t^2\) | M1 M1 A1 | |
| \(y = x - \frac{g}{u^2}x^2\) | ||
| Need \(15 \leq 30 - 900\frac{g}{u^2}\) | M1 A1 M1 A1 | |
| \(u \geq 60g\) | ||
| \(u \geq 24.2\) m/s\(^{-1}\) | M1 A1 M1 A1 | |
| (b) At max. height, \(u\sin 45° - gt = 0\) | M1 A1 M1 A1 | |
| \(t = 1.75\) | ||
| \(y_{\max} = 15\) | M1 A1 M1 A1 | |
| (c) When \(t = 3.5\), \(x = 60\) m | M1 A1 | |
| (d) Ball modelled as particle; constant gravity; etc. | B1 B1 | Total: 15 marks |
**(a)** $x = (u\cos 45°)t$, $y = (u\sin 45°)t - 4.9t^2$ | M1 M1 A1 |
$y = x - \frac{g}{u^2}x^2$ | |
Need $15 \leq 30 - 900\frac{g}{u^2}$ | M1 A1 M1 A1 |
$u \geq 60g$ | |
$u \geq 24.2$ m/s$^{-1}$ | M1 A1 M1 A1 |
**(b)** At max. height, $u\sin 45° - gt = 0$ | M1 A1 M1 A1 |
$t = 1.75$ | |
$y_{\max} = 15$ | M1 A1 M1 A1 |
**(c)** When $t = 3.5$, $x = 60$ m | M1 A1 |
**(d)** Ball modelled as particle; constant gravity; etc. | B1 B1 | **Total: 15 marks**A golf ball is hit with initial velocity $u$ ms$^{-1}$ at an angle of $45°$ above the horizontal. The ball passes over a building which is $15$ m tall at a distance of $30$ m horizontally from the point where the ball was hit.
\begin{enumerate}[label=(\alph*)]
\item Find the smallest possible value of $u$. [7 marks]
\end{enumerate}
When $u$ has this minimum value,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that the ball does not rise higher than the top of the building. [4 marks]
\item Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
\item Briefly describe two modelling assumptions that you have made. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q8 [15]}}