| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projectile passing through given point |
| Difficulty | Standard +0.3 This is a standard M3 projectile question requiring derivation of trajectory equation (routine application of kinematic equations), substitution of given values to form a quadratic in tan θ, solving the quadratic, and finding time of flight. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown3.02i Projectile motion: constant acceleration model |
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Could you please provide the full mark scheme content for Question 2? I'm ready to convert Unicode symbols to LaTeX notation and format it properly once you share the complete marking points.2 A projectile is fired from a point $O$ on top of a hill with initial velocity $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\theta$ above the horizontal and moves in a vertical plane. The horizontal and upward vertical distances of the projectile from $O$ are $x$ metres and $y$ metres respectively.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that, during the flight, the equation of the trajectory of the projectile is given by
$$y = x \tan \theta - \frac { g x ^ { 2 } } { 12800 } \left( 1 + \tan ^ { 2 } \theta \right)$$
\item The projectile hits a target $A$, which is 20 m vertically below $O$ and 400 m horizontally from $O$.\\
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-04_392_1031_970_460}
Show that
$$49 \tan ^ { 2 } \theta - 160 \tan \theta + 41 = 0$$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the two possible values of $\theta$. Give your answers to the nearest $0.1 ^ { \circ }$.
\item Hence find the shortest possible time of the flight of the projectile from $O$ to $A$.
\end{enumerate}\item State a necessary modelling assumption for answering part (a)(i).
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-05_2484_1709_223_153}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-07_2484_1709_223_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2010 Q2 [13]}}