| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projection from elevated point - angle above horizontal |
| Difficulty | Moderate -0.3 This is a straightforward two-part projectiles question requiring standard application of SUVAT equations with vertical motion. Part (i) involves solving a quadratic equation to find time (with the answer given), and part (ii) requires finding horizontal and vertical velocity components at impact then combining them. All steps are routine for M2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-1.5 = 29\sin30\,t - gt^2/2\) | M1 | \(5t^2 - 14.5t - 1.5 = 0\) |
| \(t = 3\) | A1 | [2 marks] AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y' = 29\sin30 - 3g\) | B1 | \(15.5\ \text{ms}^{-1}\) down, landing |
| \(v^2 = (29\cos30)^2 + 15.5^2\) or \(v^2 = 29^2 + 2g \times 1.5\) or \(\tan\theta = (3g - 29\sin30)/(29\cos30)\) | M1 | |
| \(v = 29.5\ \text{ms}^{-1}\) | A1 | |
| \(\theta = 31.7°\) with the horizontal | A1 | [4 marks] \(58.3°\) to the vertical |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-1.5 = 29\sin30\,t - gt^2/2$ | M1 | $5t^2 - 14.5t - 1.5 = 0$ |
| $t = 3$ | A1 | **[2 marks]** AG |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y' = 29\sin30 - 3g$ | B1 | $15.5\ \text{ms}^{-1}$ down, landing |
| $v^2 = (29\cos30)^2 + 15.5^2$ or $v^2 = 29^2 + 2g \times 1.5$ or $\tan\theta = (3g - 29\sin30)/(29\cos30)$ | M1 | |
| $v = 29.5\ \text{ms}^{-1}$ | A1 | |
| $\theta = 31.7°$ with the horizontal | A1 | **[4 marks]** $58.3°$ to the vertical |
---4 A small ball $B$ is projected from a point 1.5 m above horizontal ground with initial speed $29 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal.\\
(i) Show that $B$ strikes the ground 3 s after projection.\\
(ii) Find the speed and direction of motion of $B$ immediately before it strikes the ground.
\hfill \mbox{\textit{CAIE M2 2015 Q4 [6]}}