Question

When the spring is compressed to maximum, what fraction of incident kinetic energy is stored in spring?

Answer 1

Step1: Find the velocity of combine system after collision.

Momentum conservation:
\(mu=(m+3m)V\Rightarrow V=\dfrac{u}{4}\)

Step2: Find the kinetic energy is stored in spring.

Energy conservation:

\(\dfrac{1}{2} mu^2=\dfrac{1}{2}(m+3m)V^2+\dfrac{1}{2} kx^2\)

=\(\dfrac{1}{2}\times 4m \dfrac{u^2}{16}+\dfrac{1}{2} kx^2\)

\(\dfrac{1}{2} kx^2=\dfrac{3}{8} mu^2\)

Fraction of incident \(K. E.\) stored in the spring is

\(\dfrac{\dfrac{1}{2}kx^2}{\dfrac{1}{2}mu^2}=\dfrac{\dfrac{3}{8}}{\dfrac{1}{2}}=\dfrac{3}{4}\)