| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Modelling assumptions and limitations |
| Difficulty | Standard +0.3 This is a standard M2 projectile motion question with routine calculations: SUVAT for vertical drop, parametric equations for projectile motion, Cartesian conversion, and speed calculation using components. All techniques are textbook exercises requiring no novel insight, though part (e) involves multiple steps. Slightly easier than average due to straightforward application of standard methods. |
| Spec | 1.02n Sketch curves: simple equations including polynomials3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(s = \frac{1}{4}g t^2 = \frac{1}{4} \times 9 \cdot 8 \times 3 \cdot 3^2 = 53 \cdot 4\) m | M1 A1 | |
| (b) Ball, being lighter, may be affected by air resistance; include this | B1 B1 | |
| (c) \(x = (7 \cos 30°)t = \frac{7\sqrt{3}}{2}t\) | M1 A1 M1 A1 | |
| \(y = (7 \sin 30°)t - \frac{1}{2}gt^2 = \frac{7}{2}t - 4 \cdot 9t^2\) | M1 | |
| (d) \(t = \frac{7}{2\sqrt{3}}\) | M1 A1 A1 | |
| \(y = \frac{\sqrt{3}}{3} - 4 \cdot 9\left(\frac{7}{2\sqrt{3}}\right)^2 = \frac{\sqrt{3}}{3} - \frac{7}{15}x^2\) | M1 A1 A1 | |
| (e) When \(x = 10\), \(t = 1 \cdot 65\) | M1 A1 A1 | |
| \(v_x = 3 \cdot 5\sqrt{3}\), \(v_y = 3 \cdot 5 - 1 \cdot 65g = -12 \cdot 67\) | M1 A1 A1 | |
| \(v = \sqrt{(6 \cdot 062^2 + 12 \cdot 67^2)} = 14 \cdot 0\) ms\(^{-1}\) | M1 | (16 marks) |
(a) $s = \frac{1}{4}g t^2 = \frac{1}{4} \times 9 \cdot 8 \times 3 \cdot 3^2 = 53 \cdot 4$ m | M1 A1 |
(b) Ball, being lighter, may be affected by air resistance; include this | B1 B1 |
(c) $x = (7 \cos 30°)t = \frac{7\sqrt{3}}{2}t$ | M1 A1 M1 A1 |
$y = (7 \sin 30°)t - \frac{1}{2}gt^2 = \frac{7}{2}t - 4 \cdot 9t^2$ | M1 |
(d) $t = \frac{7}{2\sqrt{3}}$ | M1 A1 A1 |
$y = \frac{\sqrt{3}}{3} - 4 \cdot 9\left(\frac{7}{2\sqrt{3}}\right)^2 = \frac{\sqrt{3}}{3} - \frac{7}{15}x^2$ | M1 A1 A1 |
(e) When $x = 10$, $t = 1 \cdot 65$ | M1 A1 A1 |
$v_x = 3 \cdot 5\sqrt{3}$, $v_y = 3 \cdot 5 - 1 \cdot 65g = -12 \cdot 67$ | M1 A1 A1 |
$v = \sqrt{(6 \cdot 062^2 + 12 \cdot 67^2)} = 14 \cdot 0$ ms$^{-1}$ | M1 | (16 marks)A piece of lead and a table tennis ball are dropped together from a point $P$ near the top of the Leaning Tower of Pisa. The lead hits the ground after 3.3 seconds.
\begin{enumerate}[label=(\alph*)]
\item Calculate the height above ground from which the lead was dropped. [2 marks]
\end{enumerate}
According to a simple model, the ball hits the ground at the same time as the lead.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. [2 marks]
\end{enumerate}
The piece of lead is now thrown again from $P$, with speed 7 ms$^{-1}$ at an angle of 30° to the horizontal, as shown.
\includegraphics{figure_6}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find expressions in terms of $t$ for $x$ and $y$, the horizontal and vertical displacements respectively of the piece of lead from $P$ at time $t$ seconds after it is thrown. [4 marks]
\item Deduce that $y = \frac{\sqrt{3}}{3}x - \frac{2}{15}x^2$. [3 marks]
\item Find the speed of the piece of lead when it has travelled 10 m horizontally from $P$. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q6 [16]}}