| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 1 |
| Topic | Power and driving force |
| Type | Variable resistance: find k or constants |
| Difficulty | Standard +0.3 This is a standard mechanics question involving power, resistance, and differential equations. Part (a) is straightforward substitution at terminal velocity. Part (b) requires applying F=ma with P=Fv, which is routine for Further Maths students. Part (c) involves separating variables and integrating, which is a standard technique. The algebra is manageable and the question follows a well-established template for power-resistance problems, making it slightly easier than average for Pre-U Further Maths. |
| Spec | 6.02l Power and velocity: P = Fv6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = -60\int_3^7 \dfrac{-2v}{100 - v^2}\,\text{d}v = \left[-60\ln | 100 - v^2 | \right]_3^7\) (Limits not required) [M1A1] |
**Question 10(a):**
Tractive force $=$ Resistance at steady speed $\Rightarrow \dfrac{75}{100} = 10k \Rightarrow k = \dfrac{3}{4}$ **AG** [B1]
**Total: 1 mark**
**Question 10(b):**
$F = ma \Rightarrow \dfrac{75}{v} - \dfrac{3}{4} = 90\dfrac{\text{d}v}{\text{d}t} \Rightarrow \dfrac{25}{v} - \dfrac{1}{4}v = 30\dfrac{\text{d}v}{\text{d}t}$ **AG** (3 terms required for M1) [M1A1]
**Total: 2 marks**
**Question 10(c):**
$\int_0^t \text{d}t = \int_3^7 \dfrac{120v}{100 - v^2}\,\text{d}v$ [M1]
$t = -60\int_3^7 \dfrac{-2v}{100 - v^2}\,\text{d}v = \left[-60\ln|100 - v^2|\right]_3^7$ (Limits not required) [M1A1]
$= -60\ln 51 + 60\ln 91 = 60\ln\left(\dfrac{91}{51}\right) (= 34.7)$ seconds [M1A1]
**Total: 5 marks**10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time $t$ seconds her speed is $v \mathrm {~ms} ^ { - 1 }$ and the resistance to motion is $k v \mathrm {~N}$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Given that the steady speed at which the cyclist can move is $10 \mathrm {~ms} ^ { - 1 }$, show that $k = \frac { 3 } { 4 }$.
\item Show that
$$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
\item Find the time taken for the cyclist to accelerate from a speed of $3 \mathrm {~ms} ^ { - 1 }$ to a speed of $7 \mathrm {~ms} ^ { - 1 }$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2020 Q10 [1]}}