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 $\mathrm{X}\mathrm{m}$. The value of $\mathrm{X}$, to the nearest integer, is _______

Answer 1

Step 1: Given data

Mass of a ball $\mathrm{m}=4\mathrm{Kg}$

Velocity of a ball $\mathrm{v}=10{\mathrm{ms}}^{-1}$

Natural length of a spring$=8\mathrm{m}$

Force constant of a spring $\mathrm{k}=100{\mathrm{Nm}}^{-1}$

Let length of compressed spring$=\mathrm{X}\mathrm{m}$

Let spring is compressed by $\text{'}\mathrm{y}\text{'}\mathrm{m}$

Step 2: Calculating the length of compresses spring $\mathrm{X}$

Here the kinetic energy of a ball will get converted into the potential energy when the ball strikes the spring. Hence applying the law of conservation of energy we get:

Kinetic energy $=$ Potential energy

$$\begin{gathered} \frac{1}{2}{\mathrm{mv}}^2=\frac{1}{2}{\mathrm{ky}}^2 \\ \Rightarrow {\mathrm{mv}}^2={\mathrm{ky}}^2 \\ \Rightarrow \mathrm{y}=\sqrt{\frac{\mathrm{m}}{\mathrm{k}}{v}^2} \\ \Rightarrow \mathrm{y}=\sqrt{\frac{4}{100}{\left(10\right)}^2} \\ \Rightarrow \mathrm{y}=2\mathrm{m} \end{gathered}$$

Therefore the length of a compressed spring is given by:

$$\begin{gathered} \mathrm{Natural}\mathrm{length}-\mathrm{y} \\ =8-2 \\ =6\mathrm{m} \end{gathered}$$

Hence the length of a compressed spring is equal to $6\mathrm{m}$.