| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Air resistance kv - vertical motion |
| Difficulty | Standard +0.8 This M3 question requires setting up Newton's second law with air resistance, then manipulating the differential equation using the chain rule (dv/dt = v·dv/dx) and algebraic manipulation to reach a specific form, followed by integration involving partial fractions. While the steps are systematic for M3 students, the algebraic manipulation to the given form and subsequent integration require careful technique beyond routine application. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
4 A particle of mass $m \mathrm {~kg}$ is released from rest at a fixed point $O$ and falls vertically. The particle is subject to an upward resisting force of magnitude $0.49 m v \mathrm {~N}$ where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of the particle when it has fallen a distance of $x \mathrm {~m}$ from $O$.\\
(i) Write down a differential equation for the motion of the particle, and show that the equation can be written as $\left( \frac { 20 } { 20 - v } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.49$.\\
(ii) Hence find an expression for $x$ in terms of $v$.
\hfill \mbox{\textit{OCR M3 2008 Q4 [10]}}