| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Vector form projectile motion |
| Difficulty | Standard +0.3 This is a standard M2 projectiles question using vector notation. Part (a) requires setting up and solving a quadratic equation from the vertical motion equation (routine but multi-step), part (b) is straightforward substitution into horizontal motion, and part (c) applies standard velocity component formulas. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors3.02d Constant acceleration: SUVAT formulae3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = 5\) | B1 M1 A1 DM1 DM1 A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(30 = 2ut \Rightarrow 30 = 10u \Rightarrow u = 3\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v \approx 34.5\) (ms\(^{-1}\)) accept 35 | M1 A1 A1 DM1 A1 | (5 marks) [13 marks total] |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_B \approx 34.5\) (ms\(^{-1}\)) accept 35 | M1 A(2,1,0) DM1 A1 | (5 marks) |
**Part (a)**
$\Rightarrow 30 = 2ut$
$\uparrow -47.5 = 5ut - 4.9t^2$
$-47.5 = 75 - 4.9t^2$ eliminating $u$ or $t$
$t^2 = \frac{75 + 47.5}{4.9}$ ($= 25$)
$t = 5$ | B1 M1 A1 DM1 DM1 A1 | (6 marks)
cso
**Part (b)**
$30 = 2ut \Rightarrow 30 = 10u \Rightarrow u = 3$ | M1 A1 | (2 marks)
**Part (c)**
$\uparrow \dot{y} = 5u - 9.8t = -34$
$\Rightarrow \dot{x} = 2u = 6$
$v^2 = 6^2 + (-34)^2$
$v \approx 34.5$ (ms$^{-1}$) accept 35 | M1 A1 A1 DM1 A1 | (5 marks) [13 marks total]
M1 requires both $\dot{x}$ and $\dot{y}$
**Alternative to (c)**
$\frac{1}{2}m\dot{v}_B^2 - \frac{1}{2}m\dot{v}_A^2 = m \times g \times 47.5$ with $\dot{v}_A^2 = 6^2 + 15^2 = 261$
$\dot{v}_B^2 = 261 + 2 \times 9.8 \times 47.5$ ($= 1192$)
$v_B \approx 34.5$ (ms$^{-1}$) accept 35 | M1 A(2,1,0) DM1 A1 | (5 marks)
BEWARE: Watch out for incorrect use of $v^2 = u^2 + 2as$
---\includegraphics{figure_3}
[In this question, the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are in a vertical plane, $\mathbf{i}$ being horizontal and $\mathbf{j}$ being vertical.]
A particle $P$ is projected from the point $A$ which has position vector $47.5\mathbf{j}$ metres with respect to a fixed origin $O$. The velocity of projection of $P$ is $(2u\mathbf{i} + 5u\mathbf{j})$ m s$^{-1}$. The particle moves freely under gravity passing through the point $B$ with position vector $30\mathbf{i}$ metres, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the time taken for $P$ to move from $A$ to $B$ is 5 s. [6]
\item Find the value of $u$. [2]
\item Find the speed of $P$ at $B$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2008 Q6 [13]}}