Question

A ball with a speed of $9{\mathrm{ms}}^{-1}$ collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of ${30}^0$ with the original direction. The ratio of velocities of the balls after collision is$\mathrm{x}:\mathrm{y}$ where $\mathrm{x}$ is _____

Answer 1

Step 1: Given data and the diagram

The initial speed of a ball before collision in x-direction,${\mathrm{u}}_{\mathrm{x}}=9{\mathrm{ms}}^{-1}$

Initial speed of a ball in y-direction,${\mathrm{u}}_{\mathrm{y}}=0$

The mass of two balls is identical which is equal to, $\mathrm{m}$

After collision, the direction of each ball makes an angle of ${30}^0$ with x-axis. Hence,

Velocity of a ball after collision in y-direction, ${\mathrm{v}}_{\mathrm{y}}={\mathrm{v}}_1\sin {\left(30\right)}^0-{\mathrm{v}}_2\sin {\left(30\right)}^0$

The diagram for the situation is given below:

Step 2: Finding the ratio of velocities of a balls

Apply law of conservation of momentum in y-direction, we get

$$\begin{gathered} 0={\mathrm{mv}}_{\mathrm{y}} \\ \Rightarrow 0=\mathrm{m}({\mathrm{v}}_1\sin {\left(30\right)}^0-{\mathrm{v}}_2\sin {\left(30\right)}^0) \\ \Rightarrow 0=\frac{{\mathrm{mv}}_1}{2}-\frac{{\mathrm{mv}}_2}{2} \\ \Rightarrow {\mathrm{mv}}_1={\mathrm{mv}}_2 \\ \Rightarrow {\mathrm{v}}_1={\mathrm{v}}_2 \\ \Rightarrow \frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}=1=\frac{\mathrm{x}}{\mathrm{y}} \end{gathered}$$

Hence, the ratio of velocities of two ball is equal to $1$.