| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Basic trajectory calculations |
| Difficulty | Moderate -0.3 This is a standard M1 projectiles question covering basic trajectory calculations with given initial conditions. Parts (b)-(d) are routine applications of SUVAT equations and projectile formulas, while part (e) requires recognizing that minimum speed occurs at maximum height (horizontal component only). The 'show that' in part (b) and the straightforward structure make this slightly easier than average for A-level, though it requires competent handling of multiple standard techniques. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| A particle or no spin. No air resistance or no wind or only gravity acting | B1 | First assumption |
| B1 | Second assumption | |
| If more than 2 assumptions given, subtract one mark for each incorrect additional assumption | ||
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(0 = 25 \sin 40°t - 4.9t^2\) | M1 | Equation for time of flight |
| \(0 = t(25 \sin 40° - 4.9t)\) | A1 | Correct equation |
| \(t = 0\) or \(t = \frac{25 \sin 40°}{4.9}\) | dM1 | Solving for \(t\) |
| Time of flight \(= 3.28 \text{ s}\) | A1 | AG Correct final answer from correct working. (Verification method M1A1M1A0) |
| Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(s = 3.28 \times 25 \cos 40° = 62.8 \text{ m}\) | M1 | Finding range |
| A1 | Correct range | |
| B1 | Speed | |
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(25 \text{ ms}^{-1}\) at \(40°\) below the horizontal | B1 | Speed |
| B1 | Direction | |
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_{\min} = 25 \cos 40° = 19.2 \text{ ms}^{-1}\) | M1 | Horizontal component of velocity |
| A1 | Correct speed | |
| OR | ||
| \(v_{\min} = \frac{62.807}{3.2795} = 19.2 \text{ ms}^{-1}\) | ||
| Total: 12 marks |
**7(a)**
| A particle or no spin. No air resistance or no wind or only gravity acting | B1 | First assumption |
| | B1 | Second assumption |
| | | If more than 2 assumptions given, subtract one mark for each incorrect additional assumption |
| **Total: 2 marks** | | |
**7(b)**
| $0 = 25 \sin 40°t - 4.9t^2$ | M1 | Equation for time of flight |
| $0 = t(25 \sin 40° - 4.9t)$ | A1 | Correct equation |
| $t = 0$ or $t = \frac{25 \sin 40°}{4.9}$ | dM1 | Solving for $t$ |
| Time of flight $= 3.28 \text{ s}$ | A1 | AG Correct final answer from correct working. (Verification method M1A1M1A0) |
| **Total: 4 marks** | | |
**7(c)**
| $s = 3.28 \times 25 \cos 40° = 62.8 \text{ m}$ | M1 | Finding range |
| | A1 | Correct range |
| | B1 | Speed |
| **Total: 2 marks** | | |
**7(d)**
| $25 \text{ ms}^{-1}$ at $40°$ below the horizontal | B1 | Speed |
| | B1 | Direction |
| **Total: 2 marks** | | |
**7(e)**
| $v_{\min} = 25 \cos 40° = 19.2 \text{ ms}^{-1}$ | M1 | Horizontal component of velocity |
| | A1 | Correct speed |
| **OR** | | |
| $v_{\min} = \frac{62.807}{3.2795} = 19.2 \text{ ms}^{-1}$ | | |
| **Total: 12 marks** | | |7 An arrow is fired from a point $A$ with a velocity of $25 \mathrm {~ms} ^ { - 1 }$, at an angle of $40 ^ { \circ }$ above the horizontal. The arrow hits a target at the point $B$ which is at the same level as the point $A$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}
\begin{enumerate}[label=(\alph*)]
\item State two assumptions that you should make in order to model the motion of the arrow.\\
(2 marks)
\item Show that the time that it takes for the arrow to travel from $A$ to $B$ is 3.28 seconds, correct to three significant figures.
\item Find the distance between the points $A$ and $B$.
\item State the magnitude and direction of the velocity of the arrow when it hits the target.
\item Find the minimum speed of the arrow during its flight.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2007 Q7 [12]}}