| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2008 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Variable Force |
| Type | Limiting or terminal velocity |
| Difficulty | Challenging +1.2 This M4 question requires understanding power-force relationships (P=Fv), forming and solving a differential equation, and interpreting terminal velocity. The algebraic manipulation is moderately involved but follows standard techniques. The multi-step nature and need to connect several mechanics concepts makes it harder than average, though it's a fairly standard M4 exercise without requiring novel insight. |
| Spec | 6.02l Power and velocity: P = Fv6.06a Variable force: dv/dt or v*dv/dx methods |
2 A car of mass $m \mathrm {~kg}$ starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power $P$ watts, given by $P = m \left( k ^ { 2 } - v ^ { 2 } \right)$, where $v \mathrm {~ms} ^ { - 1 }$ is the velocity of the car and $k$ is a positive constant. The displacement from O in the direction of motion is $x \mathrm {~m}$.\\
(i) Show that $\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1$, and hence find $x$ in terms of $v$ and $k$.\\
(ii) How far does the car travel before reaching $90 \%$ of its terminal velocity?
\hfill \mbox{\textit{OCR MEI M4 2008 Q2 [12]}}