A light spring \(AB\) has natural length \(a\) and modulus of elasticity \(5mg\). The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(km\) is moving with speed \(\sqrt{4ga}\) along the horizontal surface towards \(P\) in the direction \(BA\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac{2}{3}a\). Find the value of \(k\). [6]

Question 2:
2 | At the collision of P and Q: ( m+km ) v=kmu | M1 | Momentum conserved, allow missing k on RHS.
k 4ga
So v=
( 1+k ) | A1
2
1 5mg a  mga
EPE = × × =
   
2 a 5  10  | B1
2
Loss in KE = Gain in EPE: 1 m ( k+1 ) v2 = 1 × 5mg ×   a 
2 2 a 5 | M1 | Energy equation, LHS correct, EPE dimensionally correct.
Substitute for v and rearrange to form quadratic equation in k
20k2 =1+k | M1
1
k =
4 | A1
6
Question | Answer | Marks | Guidance

A light spring $AB$ has natural length $a$ and modulus of elasticity $5mg$. The end $A$ of the spring is attached to a fixed point on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the end $B$ of the spring. The spring and particle $P$ are at rest on the surface.

Another particle $Q$ of mass $km$ is moving with speed $\sqrt{4ga}$ along the horizontal surface towards $P$ in the direction $BA$. The particles $P$ and $Q$ collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is $\frac{2}{3}a$.

Find the value of $k$. [6]

\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q2 [6]}}