Question

A block of mass m moving at a speed $\nu$ compresses a spring through a distance x before its speed is halved. Find the spring constant of the spring.

Answer 1

Let the velocity of the body at P be $\nu$.
So, the velocity of the body at Q is $\frac{\nu }{2}$.
Energy at point P = Energy at point Q



$$\begin{gathered} \mathrm{So},\frac{1}{2}m{\nu }_{\mathrm{P}}^2-\left(\frac{1}{2}\right)m{v}_{\mathrm{Q}}^2=\frac{1}{2}k{x}^2 \\ \Rightarrow \frac{1}{2}k{x}^2=\frac{1}{2}m\left({\mathrm{V}}_{\mathrm{P}}^2-{\mathrm{V}}_{\mathrm{Q}}^2\right) \\ \Rightarrow k{x}^2=m\left({\nu }^2-\frac{{\nu }^2}{4}\right) \\ \Rightarrow k{x}^2=m\frac{\left(4{\nu }^2-{\nu }^2\right)}{4} \\ \Rightarrow k=\frac{3m{v}^2}{4x^2} \end{gathered}$$