| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Distance traveled with variable force |
| Difficulty | Challenging +1.2 This is a structured M3 variable force question with clear signposting through three parts. Part (i) requires applying F=ma to obtain a differential equation (routine for M3). Part (ii) involves separating variables and integrating (standard technique), though the algebra requires care. Part (iii) is a straightforward definite integration. While it requires multiple techniques and careful manipulation, the question provides significant scaffolding and uses standard M3 methods throughout, making it moderately above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods |
4 A particle $P$ of mass 0.2 kg travels in a straight line on a horizontal surface. It passes through a point $O$ on the surface with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A resistive force of magnitude $0.2 \left( v + v ^ { 2 } \right) \mathrm { N }$ acts on $P$ in the direction opposite to its motion, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of $P$ when it is at a distance $x \mathrm {~m}$ from $O$.\\
(i) Show that $\frac { 1 } { 1 + v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1$.\\
(ii) By solving the differential equation in part (i) show that $\frac { - \mathrm { e } ^ { x } } { 3 - \mathrm { e } ^ { x } } \frac { \mathrm {~d} x } { \mathrm {~d} t } = - 1$, where $t$ s is the time taken for $P$ to travel $x \mathrm {~m}$ from $O$.\\
(iii) Hence find the value of $t$ when $x = 1$.
\hfill \mbox{\textit{OCR M3 2010 Q4 [11]}}