| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Basic trajectory calculations |
| Difficulty | Standard +0.0 Unable to rate — garbled OCR content |
| Spec | 3.02a Kinematics language: position, displacement, velocity, acceleration3.03c Newton's second law: F=ma one dimension |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r(\frac{d\theta}{dt})^2 = (2\sin 2t)(2\cos 2t) = 2\sin 4t\) | M1 | Find radial acceleration from \(r(\frac{d\theta}{dt})^2 \equiv r\omega^2\) |
| \(r(\frac{d\theta}{dt})^2 = 4a\sin^2 4t\) A.G. | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{d^2\theta}{dt^2} = 8\cos 4t = 0\) | M1 | Find \(t\) by equating \(\frac{d^2\theta}{dt^2}\) to 0 |
| \(t = \frac{\pi}{8}\) or \(0.393\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4ma\sin^2\frac{4\pi}{12} = 3ma\) and \(8ma\cos\frac{4\pi}{12} = 4ma\) | M1 | Find radial and transverse components of force |
| \(\sqrt{3^2 + 4^2}\, ma = 5ma\) | A1 | Combine to find magnitude of resultant force |
# Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r(\frac{d\theta}{dt})^2 = (2\sin 2t)(2\cos 2t) = 2\sin 4t$ | M1 | Find radial acceleration from $r(\frac{d\theta}{dt})^2 \equiv r\omega^2$ |
| $r(\frac{d\theta}{dt})^2 = 4a\sin^2 4t$ **A.G.** | A1 | |
## Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d^2\theta}{dt^2} = 8\cos 4t = 0$ | M1 | Find $t$ by equating $\frac{d^2\theta}{dt^2}$ to 0 |
| $t = \frac{\pi}{8}$ or $0.393$ | A1 | |
## Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4ma\sin^2\frac{4\pi}{12} = 3ma$ and $8ma\cos\frac{4\pi}{12} = 4ma$ | M1 | Find radial and transverse components of force |
| $\sqrt{3^2 + 4^2}\, ma = 5ma$ | A1 | Combine to find magnitude of resultant force |
---2 A particle $P$ of mass $m \mathrm {~kg}$ moves on an arc of a circle with centre $O$ and radius $a$ metres. At time $t = 0$ the particle is at the point $A$. At time $t$ seconds, angle $P O A = \sin ^ { 2 } 2 t$. Show that the radial component of the acceleration of $P$ at time $t$ seconds has magnitude $\left( 4 a \sin ^ { 2 } 4 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }$.
Find\\
(i) the value of $t$ when the transverse component of the acceleration of $P$ is first equal to zero,\\
(ii) the magnitude of the resultant force acting on $P$ when $t = \frac { 1 } { 12 } \pi$.
\hfill \mbox{\textit{CAIE FP2 2014 Q2}}