| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | Variable Force |
| Type | Power-velocity relationship |
| Difficulty | Challenging +1.2 This is a standard mechanics problem requiring application of F=ma with power-velocity relationship (P=Fv), followed by separable differential equation solving with partial fractions. While it involves multiple steps and integration techniques, it follows a well-established template for variable force problems that Further Maths students practice extensively. The 'show that' part guides students to the correct DE, reducing problem-solving demand. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
(i) Driving force $= 32000/v$
$800\dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{32000}{v} - 20v$
$\dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{1600-v^2}{40v}$ **AG**
- M1: Use $P/v$ and differential equation including $\mathrm{d}v/\mathrm{d}t$
- A1: Correctly obtain AG, need to use 800 convincingly
**[2]**
(ii) $\displaystyle\int \dfrac{40v}{1600-v^2}\,\mathrm{d}v = \int\mathrm{d}t$
$c - 20\ln(1600-v^2) = t$
$c = 20\ln 1600\ [147.56]$
$t = 20\ln\!\left(\dfrac{1600}{1600-v^2}\right)$
$v = 40\sqrt{1-e^{-t/20}}$
Tends to 40
- M1: Separate variables and attempt to integrate
- A1: Correct indefinite integral, aef
- A1: Correct value of $c$
- M1: Make $v$ subject, using e, allow $v^2$
- A1: Correct expression for $v$, aef
- B1: Conclusion, cwo but can get from implicit formula
**[6]**11 A car of mass 800 kg has a constant power output of 32 kW while travelling on a horizontal road. At time $t \mathrm {~s}$ the car's speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the resistive force has magnitude $20 v \mathrm {~N}$.\\
(i) Show that $v$ satisfies the differential equation $\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1600 - v ^ { 2 } } { 40 v }$.\\
(ii) Given that $v = 0$ when $t = 0$, solve this differential equation to find $v$ in terms of $t$. State what the solution predicts as $t$ becomes large.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q11 [6]}}