| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | SUVAT in 2D & Gravity |
| Type | Projectile motion: trajectory equation |
| Difficulty | Standard +0.3 This is a straightforward 2D projectile motion problem requiring standard SUVAT equations. Students must find maximum height (using v² = u² + 2as or time to peak) and check if stone passes through window at x=22.5m using trajectory equation. All values are given directly, requiring only systematic application of learned techniques with no novel insight or complex problem-solving. |
| Spec | 3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| At maximum height | M1 | For considering maximum height |
| \(v^2 - u^2 = 2as \Rightarrow 0^2 - 8^2 = 2 \times (-9.8) \times h\) | M1 | Use of suitable \(\text{suvat}\) equation(s) eg finding and using \(t\) for maximum height (0.816 s). Allow for use of calculus. |
| \(h = 3.265...\) | A1 | CAO but allow 3.26 as well as 3.27 |
| \((3.265... < 4)\) so the stone misses the pigeon | A1 | Dependent on previous mark |
| Alternative | ||
| Substitute \(y = 4\) in \(y = 8t - 4.9t^2\) | M1 | |
| Attempt to solve \(4.9t^2 - 8t + 4 = 0\) | M1 | |
| Discriminant (= \(64 - 4 \times 4.9 \times 4 = -14.4) < 0\) | A1 | |
| No value of \(t\) so the stone does not reach height 4 m | A1 | |
| Time to house is \(\frac{22.5}{15} = 1.5\) s | B1 | |
| Height at house \(= 8 \times 1.5 - \frac{1}{2} \times 9.8 \times 1.5^2 = 0.975\) m | B1 | Allow answers from essentially correct working that round to 0.96, 0.97 or 0.98, eg 0.96375 from \(g = 9.81\) |
| \(0.8 < 0.975 < 2.0\) so it hits the window. | B1 | A 2-sided inequality must be given, either in figures or in words. Condone \(0.8 < 0.975 < 1.2\) Dependent on previous mark |
| Answer | Mark | Guidance |
|--------|------|----------|
| At maximum height | M1 | For considering maximum height |
| $v^2 - u^2 = 2as \Rightarrow 0^2 - 8^2 = 2 \times (-9.8) \times h$ | M1 | Use of suitable $\text{suvat}$ equation(s) eg finding and using $t$ for maximum height (0.816 s). Allow for use of calculus. |
| $h = 3.265...$ | A1 | CAO but allow 3.26 as well as 3.27 |
| $(3.265... < 4)$ so the stone misses the pigeon | A1 | Dependent on previous mark |
| **Alternative** | | |
| Substitute $y = 4$ in $y = 8t - 4.9t^2$ | M1 | |
| Attempt to solve $4.9t^2 - 8t + 4 = 0$ | M1 | |
| Discriminant (= $64 - 4 \times 4.9 \times 4 = -14.4) < 0$ | A1 | |
| No value of $t$ so the stone does not reach height 4 m | A1 | |
| Time to house is $\frac{22.5}{15} = 1.5$ s | B1 | |
| Height at house $= 8 \times 1.5 - \frac{1}{2} \times 9.8 \times 1.5^2 = 0.975$ m | B1 | Allow answers from essentially correct working that round to 0.96, 0.97 or 0.98, eg 0.96375 from $g = 9.81$ |
| $0.8 < 0.975 < 2.0$ so it hits the window. | B1 | A 2-sided inequality must be given, either in figures or in words. Condone $0.8 < 0.975 < 1.2$ Dependent on previous mark |
---5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.\\
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-4_400_1221_1078_411}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\begin{itemize}
\item The stone is thrown from point O on level ground. Its initial velocity is $15 \mathrm {~ms} ^ { - 1 }$ in the horizontal direction and $8 \mathrm {~ms} ^ { - 1 }$ in the vertical direction.
\item The pigeon is at point P which is 4 m above the ground.
\item The house is 22.5 m from O .
\item The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.
\end{itemize}
Show that the stone does not reach the height of the pigeon.
Determine whether the stone hits the window.
\hfill \mbox{\textit{OCR MEI M1 2016 Q5 [7]}}