(i)[3]
While $P$ is moving and the string is stretched, show that $v\frac{dv}{dx} = 32 - 40x - 48x^2$.
(ii)[5]
The greatest value of $v$ is $4.5$. Find the initial value of $v$.
Mathematics 9709 · AS & A Level · Probability
Probability — practice question
A particle $P$ with mass $0.5\,\text{kg}$ is fired along a smooth horizontal surface towards a fixed point $A$. At the outset $P$ is at $O$ on the surface, and after projection it has moved a distance $x\,\text{m}$ from $O$ and is travelling with speed $v\,\text{m s}^{-1}$. Particle $P$ is attached to $A$ by a light elastic string whose natural length is $0.8\,\text{m}$ and whose modulus of elasticity is $16\,\text{N}$. The length $OA$ is $1.6\,\text{m}$ (see diagram). The motion of $P$ is opposed by a force of magnitude $24x^2\,\text{N}$.