| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Two possible trajectories through point |
| Difficulty | Standard +0.8 This is a multi-part projectiles question requiring algebraic manipulation to find two angles of projection from trajectory conditions, then calculating time and velocity components. While it involves several steps and simultaneous equations with the trajectory formula, the techniques are standard M2 material (trajectory equation, resolving motion) without requiring novel geometric insight or proof. The algebraic work is moderately demanding but methodical. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 14k - 0.8(1+k^2)\) and \(2a = 42k - 7.2(1+k^2)\) | M1 | Creates 2 simultaneous equations |
| \(42k - 7.2(1+k^2) = 2[14k - 0.8(1+k^2)]\) | M1 | Creates a single equation in \(k\) |
| \(k = 1/2\) and \(2\) | B1 | Both values |
| \(\theta = \tan^{-1}k\) | M1 | With 1 of the candidates' value of \(k\) |
| \(\theta = 63.435\) | A1 AG | Total: 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(t = 14/(35\cos63.435)\) | M1 | |
| \(t\,(= 0.89442...) = 0.894\) s | A1 | Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(V_v = 35\sin63.4 - g[42/(35\cos63.4)]\), \(\tan\alpha = 4.495/(35\cos63.4)\) | M1 | \(V_v = 4.495\) |
| \(\alpha = 15.9°\) above the horizontal | A1 | Accept \(16(.0)°\) |
| \(V^2 = 4.495^2 + (35\cos63.4)^2\) | M1 | |
| \(V = 16.3\text{ ms}^{-1}\) | A1 | Total: 4 marks |
| OR | ||
| \(2a = 48\), \(V^2 = 35^2 - 2g \times 48\) | M1 | \(42 \times 2 - 7.2(1+2^2)\) |
| \(V = 16.3\text{ ms}^{-1}\) | A1 | |
| \(\cos\alpha = 35\cos63.435/16.3\) | M1 | |
| \(\alpha = 15.9°\) | A1 | Total: 4 marks |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 14k - 0.8(1+k^2)$ and $2a = 42k - 7.2(1+k^2)$ | M1 | Creates 2 simultaneous equations |
| $42k - 7.2(1+k^2) = 2[14k - 0.8(1+k^2)]$ | M1 | Creates a single equation in $k$ |
| $k = 1/2$ and $2$ | B1 | Both values |
| $\theta = \tan^{-1}k$ | M1 | With 1 of the candidates' value of $k$ |
| $\theta = 63.435$ | A1 AG | Total: 5 marks |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $t = 14/(35\cos63.435)$ | M1 | |
| $t\,(= 0.89442...) = 0.894$ s | A1 | Total: 2 marks |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $V_v = 35\sin63.4 - g[42/(35\cos63.4)]$, $\tan\alpha = 4.495/(35\cos63.4)$ | M1 | $V_v = 4.495$ |
| $\alpha = 15.9°$ above the horizontal | A1 | Accept $16(.0)°$ |
| $V^2 = 4.495^2 + (35\cos63.4)^2$ | M1 | |
| $V = 16.3\text{ ms}^{-1}$ | A1 | Total: 4 marks |
| **OR** | | |
| $2a = 48$, $V^2 = 35^2 - 2g \times 48$ | M1 | $42 \times 2 - 7.2(1+2^2)$ |
| $V = 16.3\text{ ms}^{-1}$ | A1 | |
| $\cos\alpha = 35\cos63.435/16.3$ | M1 | |
| $\alpha = 15.9°$ | A1 | Total: 4 marks |7 A particle $P$ is projected with speed $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of $P$ from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively. The equation of the trajectory of $P$ is
$$y = k x - \frac { \left( 1 + k ^ { 2 } \right) x ^ { 2 } } { 245 }$$
where $k$ is a constant. $P$ passes through the points $A ( 14 , a )$ and $B ( 42,2 a )$, where $a$ is a constant.\\
(i) Calculate the two possible values of $k$ and hence show that the larger of the two possible angles of projection is $63.435 ^ { \circ }$, correct to 3 decimal places.
For the larger angle of projection, calculate\\
(ii) the time after projection when $P$ passes through $A$,\\
(iii) the speed and direction of motion of $P$ when it passes through $B$.
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{CAIE M2 2016 Q7 [11]}}