A small stone \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus \(3mg\) N and natural length \(2l\) m. The other end of the string is fixed to a point \(O\) at a height \(3l\) m above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2l\) m. The coefficient of restitution between \(P\) and the surface is \(e\). Calculate the value of \(e\). [9 marks] State one assumption (other than the string being light) that you have used in your solution. [1 mark]

(a) $u = $ speed before impact. Energy: $3mgl = \frac{3mg}{4\ell}P + \frac{1}{2}mu^2$ (1) | M1 A1 A1 |

$eu = $ speed after impact. Energy: $2mgl = \frac{3mg}{4\ell}P + \frac{1}{2}me^2u^2$ (2) | B1 M1 A1 |

(1) gives $u^2 = \frac{2g\ell}{}, so $\frac{s}{4} = \frac{9e^2}{4}$ | M1 A1 A1 |

$9e^2 = 5$ | M1 A1 A1 |

$e = \frac{1}{3}\sqrt{5}$ | M1 A1 A1 |

(b) Assumption: $P$ is a particle; no air resistance | B1 |

| **Total: 10 marks** |

A small stone $P$ of mass $m$ kg is attached to one end of a light elastic string of modulus $3mg$ N and natural length $2l$ m. The other end of the string is fixed to a point $O$ at a height $3l$ m above a horizontal surface. $P$ is released from rest at $O$; it hits the surface and rebounds to a height of $2l$ m. The coefficient of restitution between $P$ and the surface is $e$.

Calculate the value of $e$. [9 marks]

State one assumption (other than the string being light) that you have used in your solution. [1 mark]

\hfill \mbox{\textit{Edexcel M3  Q4 [10]}}