| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projectile on inclined plane |
| Difficulty | Challenging +1.8 This is a challenging M3 projectile motion problem requiring coordinate transformation to an inclined plane, finding impact time by solving simultaneous equations involving perpendicular distance, resolving velocities in non-standard directions, and analyzing rebound conditions with coefficient of restitution. The multi-step nature, geometric complexity, and requirement to derive an inequality for the rebound condition place it well above average difficulty, though it follows standard M3 techniques without requiring exceptional insight. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| \(l = 22\sin 50° \cdot t - \frac{1}{2}g\cos 20° \cdot t^2\) | M1 A1 | M1: Perpendicular eqn with correct terms. A1: Correct equation |
| \(\frac{1}{2}g\cos 20° \cdot t^2 - 22\sin 50° \cdot t - l = 0\) | dM1 | dM1: Solution of their 3-term quadratic eqn. |
| \(t = 3.7185\ldots\) or \(3.719\) | A1 | A1: CAO, AWRT 3.72 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dot{x} = 22\cos 50° - 9.8\sin 20° \cdot (3.7185\ldots)\) | M1 | M1: Parallel component of vel. with their time |
| \(\dot{x} = 1.678 \text{ ms}^{-1}\) | A1 | A1: Correct component, accept AWRT 1.68 |
| \(\dot{y} = 22\sin 50° - 9.8\cos 20° \cdot (3.7185\ldots)\) | M1 | M1: Perpendicular component of vel. with their time |
| \(\dot{y} = 17.39 \text{ ms}^{-1}\) | A1 | A1: Correct component, accept AWRT -17.4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\gamma < 90° - 20°\) | B1 | B1: Seeing 90-20 or 70 |
| \(\frac{17.39c}{1.678} < \tan(90° - 20°)\) | B1F M1 | B1F: Multiplying their vertical component by \(c\). M1: Correct inequality |
| \(e < 0.265\) | A1 | A1: CAO, accept AWRT 0.265 |
**(a)(i)**
| $l = 22\sin 50° \cdot t - \frac{1}{2}g\cos 20° \cdot t^2$ | M1 A1 | M1: Perpendicular eqn with correct terms. A1: Correct equation |
| $\frac{1}{2}g\cos 20° \cdot t^2 - 22\sin 50° \cdot t - l = 0$ | dM1 | dM1: Solution of their 3-term quadratic eqn. |
| $t = 3.7185\ldots$ or $3.719$ | A1 | A1: CAO, AWRT 3.72 |
**(ii)**
| $\dot{x} = 22\cos 50° - 9.8\sin 20° \cdot (3.7185\ldots)$ | M1 | M1: Parallel component of vel. with their time |
| $\dot{x} = 1.678 \text{ ms}^{-1}$ | A1 | A1: Correct component, accept AWRT 1.68 |
| $\dot{y} = 22\sin 50° - 9.8\cos 20° \cdot (3.7185\ldots)$ | M1 | M1: Perpendicular component of vel. with their time |
| $\dot{y} = 17.39 \text{ ms}^{-1}$ | A1 | A1: Correct component, accept AWRT -17.4 |
**(b)**
| $\gamma < 90° - 20°$ | B1 | B1: Seeing 90-20 or 70 |
| $\frac{17.39c}{1.678} < \tan(90° - 20°)$ | B1F M1 | B1F: Multiplying their vertical component by $c$. M1: Correct inequality |
| $e < 0.265$ | A1 | A1: CAO, accept AWRT 0.265 |
**Total: 12 marks**
---A ball is projected from a point $O$ above a smooth plane which is inclined at an angle of $20°$ to the horizontal. The point $O$ is at a perpendicular distance of $1$ m from the inclined plane. The ball is projected with velocity $22 \text{ m s}^{-1}$ at an angle of $70°$ above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point $A$.
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the time taken by the ball to travel from $O$ to $A$. [4 marks]
\item Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at $A$. [4 marks]
\end{enumerate}
\item After striking $A$, the ball rebounds and strikes the plane for a second time at a point further up than $A$.
The coefficient of restitution between the ball and the inclined plane is $e$.
Show that $e < k$, where $k$ is a constant to be determined. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2016 Q5 [12]}}