A particle $P$ with mass $0.5\,\text{kg}$ is connected to a fixed point $O$ by a light elastic string whose natural length is $1\,\text{m}$ and whose modulus of elasticity is $16\,\text{N}$. Particle $P$ is projected vertically upwards from $O$ at speed $6\,\text{m s}^{-1}$. When $P$ is at displacement $x\,\text{m}$ above $O$, a resisting force of magnitude $0.1x^2\,\text{N}$ acts on it. After projection, the upward speed of $P$ is $v\,\text{m s}^{-1}$.
(i)[2]
Show that, before the string is taut, $\frac{dv}{dx} = -10 - 0.2x^2$.
(ii)[4]
Find the velocity of $P$ at the instant the string becomes taut.
(iii)[2]
Find an expression for the acceleration of $P$ while it is moving upwards after the string becomes taut.
(iv)[4]
Verify that $P$ comes to instantaneous rest before the extension of the string is $0.5\,\text{m}$.
(c(iii))[2]
Find an expression for the acceleration of $P$ while it is moving upwards after the string becomes taut.
(c(iv))[4]
Verify that $P$ comes to instantaneous rest before the extension of the string is $0.5\,\text{m}$.
Worked solution & mark scheme
This 18-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Apply Newton’s Second Law: $0.5v\frac{dv}{dx} = -0.5g - 0.1x^2$” …