Standard +0.8 This is a standard M3 variable force problem requiring the v dv/dx = a technique and integration with a quadratic resistance term. While the setup is routine for M3 students, the algebraic manipulation and integration of 1/(constant + v²) requires careful execution and knowledge of arctan integration, placing it moderately above average difficulty.
3 An aircraft of mass 80000 kg travelling at \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) touches down on a straight horizontal runway. It is brought to rest by braking and resistive forces which together are modelled by a horizontal force of magnitude ( \(27000 + 50 v ^ { 2 }\) ) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the aircraft. Find the distance travelled by the aircraft between touching down and coming to rest.
# Question 3 (first):
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** $A$ and $B$ move off $\perp$ and $\|$ resp. to line of centres | M1 | For correct directions of motion after impact |
| $2mv_B = mu\cos\theta$ | A1 | For correct momentum equation |
| $v_B = eu\cos\theta$ | A1 | For correct restitution equation |
| Hence $e = 0.5$ | A1 **[4]** | For correct answer $0.5$ |
| **(ii)** $v_A = u\sin\theta$ | B1 | For correct equation |
| Hence $v_A = v_B \Rightarrow u\sin\theta = 0.5u\cos\theta$ | M1 | For forming the relevant equation for $\theta$ |
| So $\theta = \tan^{-1}0.5 \approx 26.6°$ | A1 **[3]** | For correct value $26.6$ |
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# Question 3 (second):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $80000v\frac{dv}{dx} = -(27000 + 50v^2)$ | M1 | For using Newton II to form a DE |
| | A1 | For correct equation including $v\frac{dv}{dx}$ |
| Hence $x = -\int\frac{1600v}{540+v^2}\,dv$ | M1 | For separation of variables |
| $= -800\ln(540+v^2) + k$ | M1 | For logarithmic form of integral |
| | A1$\checkmark$ | For correct integration of $\frac{av}{b+cv^2}$ |
| $v=90$ when $x=0 \Rightarrow k = 800\ln 8640$ | M1 | For use of initial condition to find $k$ |
| Hence when $v=0$, $x = 800\ln 16$ | M1 | For evaluation of required distance |
| So distance is $2220$ m approximately | A1 **[8]** | For correct value $2220$ |
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3 An aircraft of mass 80000 kg travelling at $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ touches down on a straight horizontal runway. It is brought to rest by braking and resistive forces which together are modelled by a horizontal force of magnitude ( $27000 + 50 v ^ { 2 }$ ) newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the aircraft. Find the distance travelled by the aircraft between touching down and coming to rest.
\hfill \mbox{\textit{OCR M3 Q3 [8]}}