| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Power and driving force |
| Type | Maximum speed on incline vs horizontal |
| Difficulty | Standard +0.3 This is a straightforward M2 work-energy-power question requiring standard application of P=Fv at maximum speed (where acceleration=0). Part (a) involves simple algebraic manipulation to find k, while part (b) adds a component of weight down the slope but follows the same method. The question is slightly above average difficulty due to the two-part structure and inclusion of an inclined plane, but requires no novel insight—just systematic application of familiar mechanics principles. |
| Spec | 6.02l Power and velocity: P = Fv |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(R \propto v\) \(\therefore R = kv\) where \(k\) is a constant | M1 |
| \(\frac{dv}{dt} - R = 0\) \(\therefore \frac{90000}{m} - 50k = 0\) | M1 A1 | |
| \(k = 36\) \(\therefore R = 36v\) | A1 | |
| (b) | \(\frac{dv}{dt} - R - mg\sin\theta = 0\) \(\therefore \frac{90000}{m} - 36v - 1200(9.8)\frac{1}{3} = 0\) | M1 A1 |
| \(90000 - 36v^2 - 840v = 0\) \(\therefore 3v^2 + 70v - 7500 = 0\) | M1 A1 | |
| Quad. form. giving \(v = 39.7\text{ ms}^{-1}\) (3sf) (clearly 63.0 not suitable) | M1 A1 | (10 marks total) |
(a) | $R \propto v$ $\therefore R = kv$ where $k$ is a constant | M1 |
| $\frac{dv}{dt} - R = 0$ $\therefore \frac{90000}{m} - 50k = 0$ | M1 A1 |
| $k = 36$ $\therefore R = 36v$ | A1 |
(b) | $\frac{dv}{dt} - R - mg\sin\theta = 0$ $\therefore \frac{90000}{m} - 36v - 1200(9.8)\frac{1}{3} = 0$ | M1 A1 |
| $90000 - 36v^2 - 840v = 0$ $\therefore 3v^2 + 70v - 7500 = 0$ | M1 A1 |
| Quad. form. giving $v = 39.7\text{ ms}^{-1}$ (3sf) (clearly 63.0 not suitable) | M1 A1 | (10 marks total)
---3. A car of mass 1200 kg experiences a resistance to motion, $R$ newtons, which is proportional to its speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the power output of the car engine is 90 kW and the car is travelling along a horizontal road, its maximum speed is $50 \mathrm {~ms} ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $R = 36 v$.
The car ascends a hill inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 1 } { 14 }$.
\item Find, correct to 3 significant figures, the maximum speed of the car up the hill assuming that the power output of the engine is unchanged.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q3 [10]}}