Question

A block strikes the free end of a horizontal spring with the other end fix, placed on a smooth surface with a speed $v$. After compressing the spring by $x$, the speed of the block reduce to half. Calculate the maximum compression of the spring.

Answer 1

Let total compression of spring =y

Applying energy conservation at P and Q,

$\frac{1}{2}m{v}^2-\frac{1}{2}m(\frac{v}{2}{)}^2=\frac{1}{2}k{x}^2$

$\frac{3m{v}^2}{4x^2}=k$.......(1)

Applying energy conservation at P and R

$\frac{1}{2}m{v}^2=\frac{1}{2}k{y}^2$.......(2)

putting (1) in (2),

$\frac{1}{2}m{v}^2=\frac{1}{2}\frac{3m{v}^2}{4x^2}{y}^2$

$y^2=\frac{4}{3}{x}^2$

$y=\frac{2x}{\sqrt{3}}$