A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x\) m from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac{8}{x}\) N. When \(x = 2\), the speed of \(P\) is 4 ms\(^{-1}\). Find the speed of \(P\) when it is 0.5 m from \(O\). [8 marks]

$F = ma$: $0.5v \frac{dv}{dx} = -\frac{8}{x^2}$ | B1 M1 |
$\int v \, dv = -\int \frac{16}{x^2} \, dx$ | |
$\frac{v^2}{2} = \frac{16}{x} + c$ with $x = 2, y = 4$: $8 = 8 + c$ | A1 M1 A1 A1 |
$c = 0$ | |
$v^2 = \frac{32}{x}$ | |
When $x = 0.5$, $v^2 = 64$ | M1 A1 |
$|v| = 8 \, \text{ms}^{-1}$ | |
| | **Total: 8 marks** |

A particle $P$ of mass 0.5 kg moves along a straight line. When $P$ is at a distance $x$ m from a fixed point $O$ on the line, the force acting on it is directed towards $O$ and has magnitude $\frac{8}{x}$ N.

When $x = 2$, the speed of $P$ is 4 ms$^{-1}$.

Find the speed of $P$ when it is 0.5 m from $O$. [8 marks]

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