Edexcel M4 — Question 4 10 marks

Exam BoardEdexcel
ModuleM4 (Mechanics 4)
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVariable Force
TypeLimiting or terminal velocity
DifficultyStandard +0.8 This is a standard M4 differential equations problem requiring setup of F=ma with air resistance, separation of variables, and integration to find time. While it involves multiple steps (forming equation, using terminal velocity condition, integrating, applying limits), it follows a well-established method taught in M4 with no novel insight required. The 10 marks reflect the working length rather than exceptional difficulty, placing it moderately above average.
Spec6.06a Variable force: dv/dt or v*dv/dx methods
4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(mg - 100k = 0\) at terminal speedM1
\(k = \dfrac{mg}{100}\)A1
\(mg - \dfrac{mg}{100}v = m\dfrac{\mathrm{d}v}{\mathrm{d}t}\)M1 A1 A1
\(\int \mathrm{d}t = \dfrac{100}{g}\int\dfrac{\mathrm{d}v}{100-v}\)M1
\(T = \dfrac{100}{g}\Big[-\ln(100-v)\Big]_{60}^{0}\)A1 A1 A1 for limits
\(= \dfrac{100}{g}\ln\!\left(\dfrac{100}{40}\right)\)M1
\(= 9.35\text{ s}\) (3 s.f.)A1 
# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $mg - 100k = 0$ at terminal speed | M1 | |
| $k = \dfrac{mg}{100}$ | A1 | |
| $mg - \dfrac{mg}{100}v = m\dfrac{\mathrm{d}v}{\mathrm{d}t}$ | M1 A1 A1 | |
| $\int \mathrm{d}t = \dfrac{100}{g}\int\dfrac{\mathrm{d}v}{100-v}$ | M1 | |
| $T = \dfrac{100}{g}\Big[-\ln(100-v)\Big]_{60}^{0}$ | A1 A1 | A1 for limits |
| $= \dfrac{100}{g}\ln\!\left(\dfrac{100}{40}\right)$ | M1 | |
| $= 9.35\text{ s}$ (3 s.f.) | A1 | |

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4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed.

Given that its terminal speed is $100 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the time it takes for the body to attain a speed of $60 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(10 marks)\\

\hfill \mbox{\textit{Edexcel M4  Q4 [10]}}
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