Question

Two balls are rolling on a flat smooth table. One ball has velocity components $\sqrt{3}\hat{j}$ and $\hat{i}$ while the other has components $2\hat{i}$ and $2\hat{j}$. If both start moving simultaneously from the same point, the angle between their paths is:

• A. ${15}^{\circ }$
• B. ${30}^{\circ }$
• C. ${45}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is A ${15}^{\circ }$

Here, $\overrightarrow{v_1}=\hat{i}+\sqrt{3}\hat{j}$ and $\overrightarrow{v_2}=2\hat{i}+2\hat{j}$
So, $|\overrightarrow{v_1}|=\sqrt{1^2+(\sqrt{3}{)}^2}=2$ and $|\overrightarrow{v_2}|=\sqrt{2^2+2^2}=2\sqrt{2}$
If $\theta$ be the angle between paths,
$\overrightarrow{v_1}.\overrightarrow{v_2}=|\overrightarrow{v_1}||\overrightarrow{v_2}|\cos \theta$
$(\hat{i}+\sqrt{3}\hat{j}).(2\hat{i}+2\hat{j})=2\left(2\sqrt{2}\right)\cos \theta$
$2+2\sqrt{3}=4\sqrt{2}\cos \theta$
$\cos \theta =0.968$
$\theta ={15}^o$