Question

A ball of mass $10\mathrm{Kg}$ moving with a velocity $10\sqrt{3}{\mathrm{ms}}^{-1}$ along the x-axis, hits another ball of mass $20\mathrm{Kg}$ which is at rest. After the collision, the first ball comes to rest while the second ball disintegrates into two equal pieces. One-piece starts moving along the y-axis with a speed of $10{\mathrm{ms}}^{-1}$. The second piece starts moving at an angle of ${30}^0$ with respect to the x-axis. The velocity of the ball moving at ${30}^0$ with x-axis is ${\mathrm{v}}_2{\mathrm{ms}}^{-1}$.The configuration of pieces after collision is shown in the figure below The value of ${\mathrm{v}}_2$ to the nearest integer is ______.

Answer 1

Step 1: Given data and drawing the diagram

We are given with the following data:

Before collision we have:

Mass of first ball, ${\mathrm{m}}_1=10\mathrm{Kg}$

Initial velocity of first ball, ${\mathrm{u}}_1=10\sqrt{3}{\mathrm{ms}}^{-1}$

Mass of second ball, ${\mathrm{m}}_2=20\mathrm{Kg}$

Initial velocity of second ball, ${\mathrm{u}}_2=0$

After collision the second ball disintegrates into two equal pieces, one piece moves along x-axis and another piece moves along y-axis.

So here we will consider, that piece which moves along x-axis to calculate the required velocity.

So after collision we have:

$$\begin{gathered} {\mathrm{v}}_1=0 \\ {\mathrm{v}}_2=\mathrm{velocity}\mathrm{of}\mathrm{second}\mathrm{ball} \\ \mathrm{\theta }={30}^0 \\ {\mathrm{v}}_2\cos {\left(30\right)}^0=\mathrm{velocity}\mathrm{of}\mathrm{second}\mathrm{ball}\mathrm{along}\mathrm{x}-\mathrm{axis}. \end{gathered}$$

Before collision:

After collision:

Step 2: Calculating the velocity $\text{'}{\mathrm{v}}_2\text{'}$

Applying the law of conservation of momentum along x-axis, we get

$$\begin{gathered} {\mathrm{m}}_1{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{u}}_2={\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2 \\ \Rightarrow 10\times 10\sqrt{3}+20\times 0=20\times 0+10\times {\mathrm{v}}_2\cos {\left(30\right)}^0 \\ \Rightarrow 100\sqrt{3}=10{\mathrm{v}}_2\cos {\left(30\right)}^0 \\ \Rightarrow {\mathrm{v}}_2\cos {\left(30\right)}^0=\frac{100\sqrt{3}}{10} \\ \Rightarrow {\mathrm{v}}_2=\frac{10\sqrt{3}}{\cos {\left(30\right)}^0}=\frac{10\sqrt{3}}{{\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \\ \Rightarrow {\mathrm{v}}_2=20{\mathrm{ms}}^{-1} \end{gathered}$$

Hence, the velocity $\text{'}{\mathrm{v}}_2\text{'}$ is equal to $20{\mathrm{ms}}^{-1}$.