Edexcel M4 2003 January — Question 3 11 marks

Exam BoardEdexcel
ModuleM4 (Mechanics 4)
Year2003
SessionJanuary
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVariable Force
TypeAir resistance kv² - falling from rest or projected downward
DifficultyChallenging +1.2 This is a standard M4 resistive forces question requiring integration of the equation of motion with v² resistance. While it involves multiple steps (setting up F=ma, separating variables, integrating, and solving for v), the method is a textbook application of techniques taught explicitly in M4. The 11 marks reflect length rather than conceptual difficulty—students who have practiced this topic will recognize the standard approach immediately.
Spec3.02h Motion under gravity: vector form6.06a Variable force: dv/dt or v*dv/dx methods
A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(mkv^2\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. [11]
AnswerMarks Guidance
ContentMarks Notes
\(\downarrow \quad mg - mkv^2 = ma\)M1 A1
\(g - kv^2 = v\frac{dv}{dx}\)M1
\(x = \int \frac{v}{g-kv^2} dv\)M1
\(x = -\frac{1}{2k} \ln \g - kv^2\ + c\)
\(x = 0, v = 0 \Rightarrow 0 = -\frac{1}{2k} + c\)M1
\(x = \frac{1}{2k}\ln\left\\frac{g}{g-kv^2}\right\ \)
\(e^{2kx} = \frac{g}{g-kv^2}\)A1
\(kv^2 = g(1-e^{-2kx})\)M1
\(v = \sqrt{\frac{g}{k}(1-e^{-2kx})}\)A1 must use \(D\)
(11 marks) 
| Content | Marks | Notes |
|---------|-------|-------|
| $\downarrow \quad mg - mkv^2 = ma$ | M1 A1 | |
| $g - kv^2 = v\frac{dv}{dx}$ | M1 | |
| $x = \int \frac{v}{g-kv^2} dv$ | M1 | |
| $x = -\frac{1}{2k} \ln \|g - kv^2\| + c$ | M1 A1 | |
| $x = 0, v = 0 \Rightarrow 0 = -\frac{1}{2k} + c$ | M1 | |
| $x = \frac{1}{2k}\ln\left\|\frac{g}{g-kv^2}\right\|$ | | |
| $e^{2kx} = \frac{g}{g-kv^2}$ | A1 | |
| $kv^2 = g(1-e^{-2kx})$ | M1 | |
| $v = \sqrt{\frac{g}{k}(1-e^{-2kx})}$ | A1 | must use $D$ |
| | **(11 marks)** | |

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A small pebble of mass $m$ is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is $v$ the magnitude of the resistance due to the liquid is modelled as $mkv^2$, where $k$ is a positive constant.

Find the speed of the pebble after it has fallen a distance $D$ through the liquid.
[11]

\hfill \mbox{\textit{Edexcel M4 2003 Q3 [11]}}
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