| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 15 |
| 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 with straightforward vector manipulation. Parts (a)-(c) involve routine calculations: finding speed from components, verifying a given equation using kinematic equations, and solving a quadratic. Part (d) requires finding maximum height by differentiation, which is slightly beyond basic projectiles but still a standard technique. The question is slightly easier than average due to its structured nature and clear signposting. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(12.5\) ms\(^{-1}\) \((7, 24, 25\,\triangle)\) | M1 A1 | |
| (b) \(x = 3.5t, y = 12t - \frac{1}{2}g t^2 = 12t - 4.9t^2\) | B1 M1 A1 | |
| hence \(\mathbf{r} = 3.5t\mathbf{i} + (12t - 4.9t^2)\mathbf{j}\) | A1 | |
| (c) When \(y = 4.4, 4.9t^2 - 12t + 4.4 = 0\) \(\quad t = (12 \pm 7.6)/9.8\) | M1 A1 | |
| \(t = 0.449\) or \(t = 2\) \(\quad x = 1.57\) m, \(x = 7\) m | M1 A1 A1 | |
| (d) \(\mathbf{v} = 3.5\mathbf{i} + (12 - 3t^2)\mathbf{j}\) \(\quad\) When \(y\)-component of \(\mathbf{v}\) is \(0, t = 2\) | B1 M1 A1 | |
| Then \(\mathbf{r} = 7\mathbf{i} + 16\mathbf{j}\) | A1 | Total: 15 marks |
**(a)** $12.5$ ms$^{-1}$ $(7, 24, 25\,\triangle)$ | M1 A1 |
**(b)** $x = 3.5t, y = 12t - \frac{1}{2}g t^2 = 12t - 4.9t^2$ | B1 M1 A1 |
hence $\mathbf{r} = 3.5t\mathbf{i} + (12t - 4.9t^2)\mathbf{j}$ | A1 |
**(c)** When $y = 4.4, 4.9t^2 - 12t + 4.4 = 0$ $\quad t = (12 \pm 7.6)/9.8$ | M1 A1 |
$t = 0.449$ or $t = 2$ $\quad x = 1.57$ m, $x = 7$ m | M1 A1 A1 |
**(d)** $\mathbf{v} = 3.5\mathbf{i} + (12 - 3t^2)\mathbf{j}$ $\quad$ When $y$-component of $\mathbf{v}$ is $0, t = 2$ | B1 M1 A1 |
Then $\mathbf{r} = 7\mathbf{i} + 16\mathbf{j}$ | A1 | Total: 15 marks8. A particle $P$ is projected from a point $O$ with initial velocity $( 3 \cdot 5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and moves under gravity. $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the horizontal and vertical directions respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the initial speed of $P$.
\item Show that the position vector $\mathbf { r } \mathbf { m }$ of $P$ at time $t$ seconds after projection is given by
$$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - 4 \cdot 9 t ^ { 2 } \right) \mathbf { j } .$$
\item Find the horizontal distance of $P$ from $O$ at each of the times when it is 4.4 m vertically above the level of $O$.
In a refined model of the motion of $P$, the position vector of $P$ at time $t$ seconds is taken to be
$$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - t ^ { 3 } \right) \mathbf { j } \mathbf { ~ m } .$$
\item Using this model, find the position vector of the highest point reached by $P$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q8 [15]}}