OCR MEI M4 2007 June — Question 4 24 marks

Exam BoardOCR MEI
ModuleM4 (Mechanics 4)
Year2007
SessionJune
Marks24
PaperDownload PDF ↗
TopicVariable Force
TypeVariable force (velocity v) - use v dv/dx
DifficultyChallenging +1.8 This M4 question requires multiple differential equation formulations and solutions using v dv/dx and dv/dt methods, integration of exponential functions, and connecting force-work-impulse concepts. While the techniques are standard for Further Maths Mechanics 4, the multi-part structure, algebraic manipulation of exponentials, and need to apply different approaches (energy method vs time-based) across four connected parts makes this significantly harder than typical A-level questions.
Spec6.02b Calculate work: constant force, resolved component6.03f Impulse-momentum: relation6.06a Variable force: dv/dt or v*dv/dx methods
4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) under the action of a force \(F \mathrm {~N}\), where \(F = 2 - 8 v ^ { 2 }\). The displacement of the particle from O at time \(t\) seconds is \(x \mathrm {~m}\).
  1. Formulate and solve a differential equation to show that \(v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)\).
  2. Hence express \(F\) in terms of \(x\) and find, by integration, the work done in the first 2 m of the motion.
  3. Formulate and solve a differential equation to show that \(v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)\).
  4. Calculate \(v\) when \(t = 1\) and when \(t = 2\), giving your answers to four significant figures. Hence find the impulse of the force \(F\) over the interval \(1 \leqslant t \leqslant 2\).
4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ under the action of a force $F \mathrm {~N}$, where $F = 2 - 8 v ^ { 2 }$. The displacement of the particle from O at time $t$ seconds is $x \mathrm {~m}$.\\
(i) Formulate and solve a differential equation to show that $v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)$.\\
(ii) Hence express $F$ in terms of $x$ and find, by integration, the work done in the first 2 m of the motion.\\
(iii) Formulate and solve a differential equation to show that $v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)$.\\
(iv) Calculate $v$ when $t = 1$ and when $t = 2$, giving your answers to four significant figures. Hence find the impulse of the force $F$ over the interval $1 \leqslant t \leqslant 2$.

\hfill \mbox{\textit{OCR MEI M4 2007 Q4 [24]}}
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