Question:
A ball of mass $4 \mathrm{~kg}$, moving with a velocity of $10 \mathrm{~ms}^{-1}$, collides with a spring of length $8 \mathrm{~m}$ and force constant $100 \mathrm{Nm}^{-1}$. The length of the compressed spring is $x \mathrm{~m}$. The value of $x$, to the nearest integer, is______.
Solution:
Let's say the compression in the spring by : y. So, by work energy theorem we have
$\Rightarrow \frac{1}{2} \mathrm{mv}=\frac{1}{2} \mathrm{ky}^{2}$
$\Rightarrow y=\sqrt{\frac{m}{k}} \cdot v$
$\Rightarrow y=\sqrt{\frac{4}{100}} \times 10$
$\Rightarrow y=2 m$
$\Rightarrow$ final length of spring
$=8-2=6 m$