Mathematics 9709 · AS & A Level · Representation of data
Representation of data — practice question
Particle $P$, with mass $0.4\,\text{kg}$, is launched horizontally on a smooth horizontal plane from point $O$. After projection, its velocity is $v\,\text{m s}^{-1}$ and its displacement from $O$ is $x\,\text{m}$. A force of magnitude $8x\,\text{N}$ acting away from $O$ acts on $P$, while another force of magnitude $(2e^{-x}+4)\,\text{N}$ acts opposite to the motion of $P$. One end of a light elastic string with natural length $0.5\,\text{m}$ is fixed to $O$, and the other end is attached to $P$.
(i)[2]
Show that $v\frac{dv}{dx} = 20x - 10 - 5e^{-x}$ while the elastic string has not yet become taut.
(ii)[3]
Given that the initial velocity of $P$ is $6\,\text{m s}^{-1}$, find $v$ when the string first becomes taut.
(iii)[2]
When the string is taut, the acceleration of $P$ is proportional to $e^{-x}$. Find the modulus of elasticity of the string.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use Newton’s Second Law horizontally to obtain $0.4a = 8x - (2e^{-x}+4)$ for $P$.” …