| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable force (position x) - find velocity |
| Difficulty | Standard +0.3 This is a straightforward application of the work-energy theorem with a variable force. Part (i) requires setting up ∫F dx = ΔKE with clear boundary conditions, and part (ii) involves solving a simple quadratic. The force law is linear and the integration is routine (∫x dx), making this slightly easier than average for A-level mechanics questions involving variable forces. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \((i)\) \(0.2dv/dx = -0.4x\) | M1 | Newton's Second Law, – sign essential |
| \(v^2/2 = -2x^2/2 (+c)\) | A1 | Accept uncancelled form |
| \(0 = -2 \times 2.5^2/2 + c \to c = 6.25\) | M1 | |
| \(KE = 0.2 \times 6.25 = 1.25\) J | A1 [4] | \(v = 3.54\) ms\(^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(2^2/2 = -2x^2/2 + 6.25\) | M1 | \(v = 2\) in accurate integral attempt at limits or finding arbitrary constant e.g. in (i) |
| \(x = 2.06\) | A1 [2] | [6] |
**(i)**
| $(i)$ $0.2dv/dx = -0.4x$ | M1 | Newton's Second Law, – sign essential |
| $v^2/2 = -2x^2/2 (+c)$ | A1 | Accept uncancelled form |
| $0 = -2 \times 2.5^2/2 + c \to c = 6.25$ | M1 | |
| $KE = 0.2 \times 6.25 = 1.25$ J | A1 [4] | $v = 3.54$ ms$^{-1}$ |
**(ii)**
| $2^2/2 = -2x^2/2 + 6.25$ | M1 | $v = 2$ in accurate integral attempt at limits or finding arbitrary constant e.g. in (i) |
| $x = 2.06$ | A1 [2] | [6] |3 A particle $P$ of mass 0.2 kg is projected horizontally from a fixed point $O$, and moves in a straight line on a smooth horizontal surface. A force of magnitude $0.4 x \mathrm {~N}$ acts on $P$ in the direction $P O$, where $x \mathrm {~m}$ is the displacement of $P$ from $O$.\\
(i) Given that $P$ comes to instantaneous rest when $x = 2.5$, find the initial kinetic energy of $P$.\\
(ii) Find the value of $x$ on the first occasion when the speed of $P$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\hfill \mbox{\textit{CAIE M2 2012 Q3 [6]}}