| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Speed at specific time or position |
| Difficulty | Standard +0.3 This is a standard projectile motion problem requiring resolution of velocity components and use of kinematic equations. The 'show that' part guides students to find the vertical component, then they apply standard distance formulas. While it involves multiple steps and both horizontal/vertical components, the techniques are routine for M2 level with no novel problem-solving required. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model6.02e Calculate KE and PE: using formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(v^2 = 17^2 - (30\cos60)^2\) | M1 | Finds vertical speed |
| \(v = -8\) | A1 [2] | – may be implied by later work |
| Answer | Marks | Guidance |
|---|---|---|
| \(-8 = 30\sin60 - gt\) | M1 | Finds relevant time |
| \(t = 3.4\) | A1 | \(3.398\) |
| \(y = [(30\sin60)^2 - 8^2]/(2g) = (30.55)\) | B1 | Or \(y = (30\sin60) \times 3.4 - g \cdot 3.4^2/2 = (30.53)\) |
| \(OP^2 = (30\cos60 \times 3.4)^2 + 30.55^2\) | M1 | Use of Pythagoras |
| \(OP = 59.4 \text{ m}\) | A1 [5] | Accept \(59.5\) |
**(i)**
$v^2 = 17^2 - (30\cos60)^2$ | M1 | Finds vertical speed
$v = -8$ | A1 [2] | – may be implied by later work
**(ii)**
$-8 = 30\sin60 - gt$ | M1 | Finds relevant time
$t = 3.4$ | A1 | $3.398$
$y = [(30\sin60)^2 - 8^2]/(2g) = (30.55)$ | B1 | Or $y = (30\sin60) \times 3.4 - g \cdot 3.4^2/2 = (30.53)$
$OP^2 = (30\cos60 \times 3.4)^2 + 30.55^2$ | M1 | Use of Pythagoras
$OP = 59.4 \text{ m}$ | A1 [5] | Accept $59.5$5 A particle $P$ is projected with speed $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. For the instant when the speed of $P$ is $17 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and increasing,\\
(i) show that the vertical component of the velocity of $P$ is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ downwards,\\
(ii) calculate the distance of $P$ from $O$.
\hfill \mbox{\textit{CAIE M2 2012 Q5 [7]}}