Mathematics 9709 · AS & A Level · Representation of data
Representation of data — practice question
A particle $P$ with mass $0.5\,\text{kg}$ is linked to a fixed point $O$ by a light elastic string whose natural length is $1\,\text{m}$ and whose modulus of elasticity is $16\,\text{N}$. The particle $P$ is then projected vertically upwards from $O$ at speed $6\,\text{m s}^{-1}$. When $P$ is a distance $x\,\text{m}$ above $O$, a resisting force of magnitude $0.1x^2\,\text{N}$ acts on it. Once projected, the upward velocity of $P$ is $v\,\text{m s}^{-1}$.
(i)[2]
Show that, while the string is still slack, $\frac{dv}{dx} = -10 - 0.2x^2$.
(ii)[4]
Find the velocity of $P$ at the moment the string first 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 $0.5\,\text{m}$.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use Newton’s Second Law: $0.5v\frac{dv}{dx} = -0.5g - 0.1x^2$” …