Question

A block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed reduced to ${\frac{1}{3}}^{rd}$ of its original value. The spring constant of the spring is $\frac{8m{v}^2}{n{x}^2}.$ Find the value of $n$

Answer 1

From Law of conservation of energy,
Gain in potential energy = Loss in kinetic energy.
From the data given in the question,
$\frac{1}{2}k{x}^2=\frac{1}{2}m(v^2-{\left(\frac{v}{3}\right)}^2)$
$\Rightarrow k{x}^2=\frac{8m{v}^2}{9}\Rightarrow k=\frac{8m{v}^2}{9x^2}$
Comparing this with spring constant given in the question $\frac{8m{v}^2}{n{x}^2}$

we get, $n=9$