A particle $P$ of mass $0.1\,\text{kg}$ travels in a straight line on a smooth horizontal surface, with its speed steadily reducing. A horizontal resisting force of magnitude $0.2e^{-x}\,\text{N}$ acts on $P$, where $x\,\text{m}$ is the displacement of $P$ from a fixed point $O$ on the line. If the displacement from $O$ is $x\,\text{m}$, the velocity of $P$ is $v\,\text{m s}^{-1}$. $P$ passes through $O$ with velocity $2.2\,\text{m s}^{-1}$.
(i)[2]
Show that $v\frac{dv}{dx} = ke^{-x}$, with $k$ a constant to be found.
(ii)[4]
Calculate the value of $x$ when the velocity of $P$ is $2\,\text{m s}^{-1}$.
(iii)[2]
Show that the speed of $P$ stays above $0.917\,\text{m s}^{-1}$, correct to $3$ significant figures.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Newton’s 2nd law used to give $0.1v\dfrac{dv}{dx} = -0.2e^{-x}$” …