Question

A block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed is reduced by a factor of $3$. Find the spring constant of the spring.

• A. $\frac{3m{v}^2}{3x^2}$
• B. $\frac{5m{v}^2}{9x^2}$
• C. $\frac{8m{v}^2}{9x^2}$
• D. $\frac{m{v}^2}{2x^2}$

Answer 1

The correct option is C $\frac{8m{v}^2}{9x^2}$

Let the velocity of the body at $A$ be $v_A=v$.

So, the velocity of the body at $B$ is $\frac{v}{3}$

Energy at point $A=$ Energy at point $B$

So, $\frac{1}{2}m{v}_A^2=\frac{1}{2}m{v}_B^2+\frac{1}{2}k{x}^2$

$\Rightarrow \frac{1}{2}k{x}^2=\frac{1}{2}m{v}_A^2-\frac{1}{2}m{v}_B^2$

$\Rightarrow k{x}^2=m(v^2-\frac{v^2}{9})$

$\Rightarrow k{x}^2=\frac{8m{v}^2}{9}$

$\Rightarrow k=\frac{8m{v}^2}{9x^2}$