| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Finding angle given constraints |
| Difficulty | Standard +0.3 This is a standard two-part projectile motion problem requiring application of kinematic equations with trigonometry. Part (i) involves setting up equations for horizontal and vertical displacement at t=0.6s using the 45° angle condition, then solving simultaneously—straightforward but requires careful algebra. Part (ii) is a direct application of finding when the velocity angle equals 45° using tan(45°)=1. Both parts use routine M2 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x = (\vcos60)0.6\) and \(y = (\vsin60)0.6 - g0.6^2/2\) | M1 | Finds both coordinates in terms of \(t = 0.6\) |
| \(\tan45 = [(\vsin60)0.6 - g0.6^2/2]/[(\vcos60)0.6]\) | DM1 | Relates coordinates and \(45°\) angle |
| \((\vsin60)0.6 - g0.6^2/2 = (\vcos60)0.6\) | A1 | |
| \(v = 8.2(0)\) m s\(^{-1}\) | AG A1 | [4] |
| (ii) \(8.2\sin60 - gt = 8.2\cos60\) | M1 | Relates velocity components and \(45°\) |
| \(T = 0.3(00)\) s | A1 | \(\tan45 = (8.2\sin60 - gt)/(8.2\cos60)\) |
| A1 | [3] |
**(i)** $x = (\vcos60)0.6$ and $y = (\vsin60)0.6 - g0.6^2/2$ | M1 | Finds both coordinates in terms of $t = 0.6$
$\tan45 = [(\vsin60)0.6 - g0.6^2/2]/[(\vcos60)0.6]$ | DM1 | Relates coordinates and $45°$ angle
$(\vsin60)0.6 - g0.6^2/2 = (\vcos60)0.6$ | A1 |
$v = 8.2(0)$ m s$^{-1}$ | AG A1 | [4]
**(ii)** $8.2\sin60 - gt = 8.2\cos60$ | M1 | Relates velocity components and $45°$
$T = 0.3(00)$ s | A1 | $\tan45 = (8.2\sin60 - gt)/(8.2\cos60)$
| | A1 | [3]\includegraphics{figure_2}
A particle $P$ is projected from a point $O$ at an angle of $60°$ above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of $P$ from $O$ is $45°$ (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Show that the speed of projection of $P$ is 8.20 m s$^{-1}$, correct to 3 significant figures. [4]
\item Calculate the time after projection when the direction of motion of $P$ is $45°$ above the horizontal. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2011 Q2 [7]}}