| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Vector form projectile motion |
| Difficulty | Moderate -0.8 This is a straightforward vector projectile question requiring only direct reading from the given equation and basic substitution. Parts (i) and (ii) involve simple extraction of values and vector arithmetic, while part (iii) is a standard elimination of parameter exercise with no conceptual challenges—all routine mechanics techniques below average A-level difficulty. |
| Spec | 1.10e Position vectors: and displacement3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (A) Height 5 m | B1 | No units required; apply ISW if incorrect units given |
| (B) \(g\) has been taken to be 10 m s\(^{-2}\) | B1 | Allow \(+10\) or \(-10\). No units required; apply ISW if incorrect units given |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Displacement is \(\begin{pmatrix}150\\80\end{pmatrix} - \begin{pmatrix}90\\80\end{pmatrix} = \begin{pmatrix}60\\0\end{pmatrix}\) | M1 | Displacement must be given as a vector. Allow a description of a vector in words. Attempts at substitution for \(t\) and subtraction of vectors must be seen |
| \(= \begin{pmatrix}60\\0\end{pmatrix}\) | A1 | Cao. If the candidate then goes on to give a non-vector answer of "60 m", apply ISW |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 30t\) | B1 | |
| \(y = 5 + 40t - 5t^2\) | B1 | |
| \(y = 5 + 40\times\left(\frac{x}{30}\right) - 5\times\left(\frac{x}{30}\right)^2\) | M1 | Attempt to eliminate \(t\) |
| \(y = 5 + \frac{4}{3}x - \frac{x^2}{180}\) | A1 | No errors |
| [4] |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| (A) Height 5 m | B1 | No units required; apply ISW if incorrect units given |
| (B) $g$ has been taken to be 10 m s$^{-2}$ | B1 | Allow $+10$ or $-10$. No units required; apply ISW if incorrect units given |
| | **[2]** | |
---
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Displacement is $\begin{pmatrix}150\\80\end{pmatrix} - \begin{pmatrix}90\\80\end{pmatrix} = \begin{pmatrix}60\\0\end{pmatrix}$ | M1 | Displacement must be given as a vector. Allow a description of a vector in words. Attempts at substitution for $t$ and subtraction of vectors must be seen |
| $= \begin{pmatrix}60\\0\end{pmatrix}$ | A1 | Cao. If the candidate then goes on to give a non-vector answer of "60 m", apply ISW |
| | **[2]** | |
---
## Question 4(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 30t$ | B1 | |
| $y = 5 + 40t - 5t^2$ | B1 | |
| $y = 5 + 40\times\left(\frac{x}{30}\right) - 5\times\left(\frac{x}{30}\right)^2$ | M1 | Attempt to eliminate $t$ |
| $y = 5 + \frac{4}{3}x - \frac{x^2}{180}$ | A1 | No errors |
| | **[4]** | |4 A projectile P travels in a vertical plane over level ground. Its position vector $\mathbf { r }$ at time $t$ seconds after projection is modelled by
$$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$
where distances are in metres and the origin is a point on the level ground.
\begin{enumerate}[label=(\roman*)]
\item Write down\\
(A) the height from which P is projected,\\
(B) the value of $g$ in this model.
\item Find the displacement of P from $t = 3$ to $t = 5$.
\item Show that the equation of the trajectory is
$$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M1 2012 Q4 [8]}}