Question

A block of mass m moving at a speed v compresses a spring through a distance x before its speed is halved, the spring constant of the spring is
A. $\dfrac{{3vm}}{{4{x^2}}}$
B.$\dfrac{{3{v^2}m}}{{4{x^2}}}$
C. $\dfrac{{3v{m^3}}}{{4{x^3}}}$
D.$\dfrac{{4{v^2}{m^2}}}{{3{x^2}}}$

Answer 1

Use the formula for the kinetic energy of an object. Also use the formula for the spring energy of the spring. Use the law of conservation of energy. According to the law of conservation of the energy, the initial kinetic energy of the block is equal to the final kinetic energy of the block and the spring energy of the spring.

Formulae used:
The kinetic energy $K$ of an object is
$K = \dfrac{1}{2}m{v^2}$ …… (1)
Here, $m$ is the mass of the object and $v$ is the velocity of the object.
The spring energy ${E_s}$ is given by
${E_s} = \dfrac{1}{2}k{x^2}$ …… (2)
Here, $k$ is the spring constant and $x$ is the displacement of the spring.

Complete step by step answer:
We have given that the block of mass $m$moving with speed $v$compresses a spring by $x$ until its speed is reduced to half of its initial speed.
Hence, the initial speed of the block is $v$ and the final speed of the block is $\dfrac{v}{2}$.
${v_i} = v$ and ${v_f} = \dfrac{v}{2}$
According to the law of conservation of energy, the initial kinetic energy ${K_i}$ of the block is equal to the sum of the final kinetic energy ${K_f}$ of the block and the gain in spring energy ${E_s}$.
${K_i} = {K_f} + {E_s}$
Substitute $\dfrac{1}{2}mv_i^2$ for ${K_i}$, $\dfrac{1}{2}mv_f^2$ for ${K_f}$ and $\dfrac{1}{2}k{x^2}$ for ${E_s}$ in the above equation.
$\dfrac{1}{2}mv_i^2 = \dfrac{1}{2}mv_f^2 + \dfrac{1}{2}k{x^2}$
Substitute $v$ for ${v_i}$ and $\dfrac{v}{2}$ for ${v_f}$ in the above equation.
$\dfrac{1}{2}m{v^2} = \dfrac{1}{2}m{\left( {\dfrac{v}{2}} \right)^2} + \dfrac{1}{2}k{x^2}$
$\Rightarrow k{x^2} = m{v^2} - m\dfrac{{{v^2}}}{4}$
$\Rightarrow k{x^2} = \dfrac{{3m{v^2}}}{4}$
$\Rightarrow k = \dfrac{{3{v^2}m}}{{4{x^2}}}$
Therefore, the spring constant of the spring is $\dfrac{{3{v^2}m}}{{4{x^2}}}$.

Hence, the correct option is B.

Note:
One can also solve the same question by considering the change in kinetic energy of the block is equal to the change in spring energy of the spring or the change in kinetic energy of the block provides the spring energy to the spring. The change in kinetic energy and the change in spring energy both will be with negative sign and hence the negative sign gets cancelled and the ultimate answer is the same.