| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Deriving trajectory equation |
| Difficulty | Standard +0.3 Part (a) is a standard derivation of the trajectory equation that appears in virtually every M3 textbook and is pure recall. Part (b) requires substituting given values into the trajectory equation to solve a quadratic for s, then finding velocity components at impact—all routine M3 techniques with no novel problem-solving required. The equal horizontal and vertical distances (both s) makes the algebra straightforward. This is slightly easier than average due to its predictability and standard methods. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=13.02i Projectile motion: constant acceleration model |
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Please share the complete mark scheme text.2 A projectile is launched from a point $O$ on top of a cliff with initial velocity $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\alpha$ and moves in a vertical plane. During the motion, the position vector of the projectile relative to the point $O$ is $( x \mathbf { i } + y \mathbf { j } )$ metres where $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical unit vectors respectively.
\begin{enumerate}[label=(\alph*)]
\item Show that, during the motion, the equation of the trajectory of the projectile is given by
$$y = x \tan \alpha - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$$
\item When $u = 21$ and $\alpha = 55 ^ { \circ }$, the projectile hits a small buoy $B$. The buoy is at a distance $s$ metres vertically below $O$ and at a distance $s$ metres horizontally from $O$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-04_601_935_964_548}
\begin{enumerate}[label=(\roman*)]
\item Find the value of $s$.
\item Find the acute angle between the velocity of the projectile and the horizontal just before the projectile hits $B$, giving your answer to the nearest degree.\\[0pt]
[5 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA M3 2015 Q2 [5]}}