CAIE M2 2017 November — Question 1 4 marks

Exam BoardCAIE
ModuleM2 (Mechanics 2)
Year2017
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVariable Force
TypeMotion with exponential force
DifficultyChallenging +1.2 This requires setting up and solving a differential equation F = ma where force depends on velocity (te^(-v) = 0.2 dv/dt), leading to a separable DE. While the exponential term adds complexity beyond standard variable force questions, the separation and integration are straightforward, and it's a single-part question with clear setup. Harder than average due to the coupled t and v dependence, but still a standard M2 technique.
Spec1.08a Fundamental theorem of calculus: integration as reverse of differentiation6.06a Variable force: dv/dt or v*dv/dx methods
1 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(t \mathrm { e } ^ { - v } \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after release. Find the velocity of \(P\) when \(t = 2\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(0.2\frac{dv}{dt} = te^{-v}\)M1 Uses Newton's Second Law to set up a differential equation. Allow \(a\) for \(dv/dt\).
\(\int e^v \, dv = 5\int t \, dt\) leading to \(e^v = 5t^2/2 (+c)\)M1 Separates the variables and integrates.
\(e^v - 1 = 2.5t^2\)A1 Substitutes \(t=0\), \(v=0\).
\(v(2) = \ln 11 = 2.4\)A1
Total4 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.2\frac{dv}{dt} = te^{-v}$ | M1 | Uses Newton's Second Law to set up a differential equation. Allow $a$ for $dv/dt$. |
| $\int e^v \, dv = 5\int t \, dt$ leading to $e^v = 5t^2/2 (+c)$ | M1 | Separates the variables and integrates. |
| $e^v - 1 = 2.5t^2$ | A1 | Substitutes $t=0$, $v=0$. |
| $v(2) = \ln 11 = 2.4$ | A1 | |
| **Total** | **4** | |

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1 A particle $P$ of mass 0.2 kg is released from rest at a point $O$ on a smooth horizontal surface. A horizontal force of magnitude $t \mathrm { e } ^ { - v } \mathrm {~N}$ directed away from $O$ acts on $P$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$ after release. Find the velocity of $P$ when $t = 2$.\\

\hfill \mbox{\textit{CAIE M2 2017 Q1 [4]}}
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Q1 4 Q2 3 Q3 6 Q4 9 Q5 8 Q6 9 Q7 11