Challenging +1.2 This is a multi-step energy problem requiring consideration of elastic potential energy, kinetic energy, and work done against friction. While it involves several components (elastic strings, friction, energy conservation), the setup is relatively standard for Further Maths mechanics with clear given parameters and a straightforward application of energy principles. The calculation is methodical rather than requiring novel insight, placing it moderately above average difficulty.
2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\).
Find the greatest extension of the string during the subsequent motion.
2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\frac { 4 } { 3 } \mathrm { mg }$. The other end of the string is attached to a fixed point $O$ on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between $P$ and the surface is $\frac { 1 } { 3 }$. The particle is projected along the surface in the direction $O P$ with a speed of $\frac { 1 } { 2 } \sqrt { \mathrm { ga } }$.
Find the greatest extension of the string during the subsequent motion.\\
\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q2 [5]}}