Question:
A block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed is halved. Find the spring constant of the spring.
Solution:
Energy at $A=$ Energy at $B$
$\frac{1}{2} \mathrm{mv}_{\mathrm{a}}=\frac{1}{2} \mathrm{mv}_{\mathrm{b}}{ }^{2}+\mathrm{k} \mathrm{x}^{2}$
$m v^{2}=\frac{m v^{2}}{4}+k x^{2}$
$m v^{2}=\frac{m v^{2}}{4}+k x^{2}$
$\mathrm{k}=\frac{3 m v^{2}}{2 x^{2}}$