3. A particle \(P\) of mass 0.5 kg moves away from the origin \(O\) along the positive \(x\)-axis under the action of a force directed towards \(O\) of magnitude \(\frac { 2 } { x ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres. When \(x = 1\), the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when its speed has been reduced to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8)

# Question 3:

| Working/Answer | Marks | Notes |
|---|---|---|
| $0.5\ddot{x} = -\dfrac{2}{x^2}$ | M1 | |
| $v\dfrac{dv}{dx} = -\dfrac{4}{x^2}$ | M1 | |
| $\int v\, dv = -\int \dfrac{4}{x^2}\, dx$ | M1 | |
| $\left[\dfrac{1}{2}v^2\right]_3^2 = \left[\dfrac{4}{x}\right]_1^d$ | M1 A1 | |
| Correct limits or constant of integration | A1 | |
| $\dfrac{9}{8} - \dfrac{9}{2} = 4\left(\dfrac{1}{d}-1\right) \Rightarrow d = \dfrac{32}{5} = 6.4$ m | M1 A1 | (8) |

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3. A particle $P$ of mass 0.5 kg moves away from the origin $O$ along the positive $x$-axis under the action of a force directed towards $O$ of magnitude $\frac { 2 } { x ^ { 2 } } \mathrm {~N}$, where $O P = x$ metres. When $x = 1$, the speed of $P$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the distance of $P$ from $O$ when its speed has been reduced to $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(8)\\

\hfill \mbox{\textit{Edexcel M3  Q3 [8]}}