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 and the second one disintegrates into two equal pieces. One of the pieces starts moving along the Y-axis at a speed of $10{\mathrm{ms}}^{-1}$. The second piece starts moving at a speed of $20{\mathrm{ms}}^{-1}$ at an angle $\mathrm{\theta }$(degree) with respect to the X-axis. The configuration of pieces after the collision is shown in the figure. The value of $\mathrm{\theta }$ to the nearest integer is ________.

Answer 1

Step 1: Given data and diagram

We are given with the data,

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

Mass of second piece of ball moving with speed $10{\mathrm{ms}}^{-1},{\mathrm{m}}_2=10\mathrm{Kg}$

Velocity of first ball, ${\mathrm{v}}_1=10\sqrt{3}{\mathrm{ms}}^{-1}$along x-axis

Velocity of second ball, ${\mathrm{v}}_2=20\cos \left(\mathrm{\theta }\right)$ along x-axis

The diagrammatic representation of the system is given below,

Step 2: Calculating the value of $\text{'}\mathrm{\theta }\text{'}$

Now applying law of conservation of linear momentum along x-axis, we have,

$$\begin{gathered} {\mathrm{m}}_1{\mathrm{v}}_1={\mathrm{m}}_2{\mathrm{v}}_2 \\ \Rightarrow 10\times 10\sqrt{3}=10\times 20\cos \left(\mathrm{\theta }\right) \\ \Rightarrow 100\sqrt{3}=200\cos \left(\mathrm{\theta }\right) \\ \Rightarrow \cos \left(\mathrm{\theta }\right)=\frac{\sqrt{3}}{2} \\ \Rightarrow \mathrm{\theta }={30}^0 \end{gathered}$$

Hence, the value of $\text{'}\mathrm{\theta }\text{'}$ nearest to integer is equal to ${30}^0$.