| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2019 |
| 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 problem involving power, resistance, and differential equations. Part (a) uses the equilibrium condition P=Fv at steady speed (straightforward). Part (b) applies F=ma with P/v - kv (routine). Part (c) requires separating variables and integrating, which is a standard Further Maths technique but involves some algebraic manipulation. The question is well-scaffolded with clear steps, making it slightly easier than average for Further Maths content. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.02l Power and velocity: P = Fv6.02m Variable force power: using scalar product6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = -60\int_3^7 \dfrac{-2v}{100 - v^2}\,dv = \left[-60\ln | 100 - v^2 | \right]_3^7\) (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)**
$F = ma \Rightarrow \dfrac{75}{v} - \dfrac{3}{4} = 90\dfrac{dv}{dt} \Rightarrow \dfrac{25}{v} - \dfrac{1}{4}v = 30\dfrac{dv}{dt}$ **AG** [M1A1]
(3 terms required for M1)
**Total: 2 marks**
**Question 10(c)**
$\int_0^t dt = \int_3^7 \dfrac{120v}{100 - v^2}\,dv$ [M1]
$t = -60\int_3^7 \dfrac{-2v}{100 - v^2}\,dv = \left[-60\ln|100 - v^2|\right]_3^7$ (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 $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, 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 2019 Q10 [1]}}