| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Deriving trajectory equation |
| Difficulty | Standard +0.3 This is a standard M2 projectiles question with a given trajectory equation. Part (i) requires simple substitution (y = -8), part (ii) involves comparing the trajectory to the standard form to extract angle and speed using routine formulas, and part (iii) uses standard kinematic equations. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-8 = 2x - x^2\) | M1 | Sub \(y = -8\) in the given equation with an attempt to solve |
| \(x = 4\) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\theta = 63.4°\) | B1 | From \(\tan^{-1} 2\) |
| \(-gx^2/(2v^2\cos^2\theta) = -x^2\) | M1 | |
| \(v = 5\) m s\(^{-1}\) | A1 (3) | Accept 4.99 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(V^2 = (v\cos\theta)^2 + (v\sin\theta)^2 + 2 \times 8g\) | M1 | |
| \(V = 13.6\) m s\(^{-1}\) | A1 (2) |
## Question 3:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-8 = 2x - x^2$ | M1 | Sub $y = -8$ in the given equation with an attempt to solve |
| $x = 4$ | A1 (2) | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\theta = 63.4°$ | B1 | From $\tan^{-1} 2$ |
| $-gx^2/(2v^2\cos^2\theta) = -x^2$ | M1 | |
| $v = 5$ m s$^{-1}$ | A1 (3) | Accept 4.99 |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V^2 = (v\cos\theta)^2 + (v\sin\theta)^2 + 2 \times 8g$ | M1 | |
| $V = 13.6$ m s$^{-1}$ | A1 (2) | |
---3 The point $O$ is 8 m above a horizontal plane. A particle $P$ is projected from $O$. After projection, 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
$$y = 2 x - x ^ { 2 }$$
(i) Find the value of $x$ for the point where $P$ strikes the plane.\\
(ii) Find the angle and speed of projection of $P$.\\
(iii) Calculate the speed of $P$ immediately before it strikes the plane.
\hfill \mbox{\textit{CAIE M2 2016 Q3 [7]}}