Answer

Verified

373.8k+ views

**Hint:**Two components of velocity, namely, x-component and y-component, are given in the problem. This implies that the balls are performing motion in the XY-plane. We shall find the relative velocity of one billiard ball with respect to the other. Further, we will use the basic knowledge of vectors to find the angle of this relative velocity and then use properties of trigonometry to find the actual angle.

**Complete answer:**

The relative velocity, ${{V}_{12}}$ of the two balls 1 and 2 is expressed as,

${{V}_{12}}={{V}_{2}}-{{V}_{1}}$

Where,

${{V}_{1}}=$ velocity of billiard ball 1

${{V}_{2}}=$ velocity of billiard ball 2

Applying the formula of relative velocity to the x-components of velocity, our equation modifies as ${{V}_{{{12}_{x}}}}={{V}_{{{2}_{x}}}}-{{V}_{{{1}_{x}}}}$.

Here, we have ${{V}_{{{1}_{x}}}}=1m{{s}^{-1}}$ and ${{V}_{{{2}_{x}}}}=2m{{s}^{-1}}$.

$\Rightarrow {{V}_{{{12}_{x}}}}=2-1$

$\Rightarrow {{V}_{{{12}_{x}}}}=1m{{s}^{-1}}$

Therefore, the x-component of relative velocity is $1m{{s}^{-1}}$.

Similarly, applying the formula of relative velocity to the x-components of velocity, our equation modifies as ${{V}_{{{12}_{y}}}}={{V}_{{{2}_{y}}}}-{{V}_{{{1}_{y}}}}$.

Here, we have ${{V}_{{{1}_{y}}}}=\sqrt{3}m{{s}^{-1}}$ and ${{V}_{{{2}_{x}}}}=2m{{s}^{-1}}$.

$\Rightarrow {{V}_{{{12}_{y}}}}=2-\sqrt{3}$

$\Rightarrow {{V}_{{{12}_{y}}}}=\left( 2-\sqrt{3} \right)m{{s}^{-1}}$

Therefore, the y-component of relative velocity is $\left( 2-\sqrt{3} \right)m{{s}^{-1}}$.

In order to find the angle, we use the property $\tan \theta =\dfrac{{{V}_{y}}}{{{V}_{x}}}$.

Here, we have ${{V}_{y}}={{V}_{{{12}_{y}}}}=\left( 2-\sqrt{3} \right)m{{s}^{-1}}$ and ${{V}_{x}}={{V}_{{{12}_{x}}}}=1m{{s}^{-1}}$. Thus, the angle, $\theta $ between their path is given as:

$\Rightarrow \tan \theta =\dfrac{2-\sqrt{3}}{1}$

We shall use the formula of $\tan 2\theta =\dfrac{2\tan \theta }{1-{{\tan }^{2}}\theta }$ and substitute the respective values.

$\Rightarrow \tan 2\theta =\dfrac{2\left( 2-\sqrt{3} \right)}{1-{{\left( 2-\sqrt{3} \right)}^{2}}}$

Opening the brackets in the denominator and writing the square of $2-\sqrt{3}$, we get

\[\begin{align}

& \Rightarrow \tan 2\theta =\dfrac{2\left( 2-\sqrt{3} \right)}{1-\left( 4+3-4\sqrt{3} \right)} \\

& \Rightarrow \tan 2\theta =\dfrac{2\left( 2-\sqrt{3} \right)}{1-7+4\sqrt{3}} \\

& \Rightarrow \tan 2\theta =\dfrac{2\left( 2-\sqrt{3} \right)}{4\sqrt{3}-6} \\

\end{align}\]

Taking $2\sqrt{3}$ common in the denominator, we get

\[\Rightarrow \tan 2\theta =\dfrac{2\left( 2-\sqrt{3} \right)}{2\sqrt{3}\left( 2-\sqrt{3} \right)}\]

Now, cancelling $\left( 2-\sqrt{3} \right)$ and 2 from numerator and denominator, we get

\[\Rightarrow \tan 2\theta =\dfrac{1}{\sqrt{3}}\]

But $\tan {{30}^{\circ }}=\dfrac{1}{\sqrt{3}}$, therefore,

\[\Rightarrow \tan 2\theta =\tan {{30}^{\circ }}\]

This implies that $2\theta ={{30}^{\circ }}$.

$\therefore \theta ={{15}^{\circ }}$

Therefore, the angle between their path is ${{15}^{\circ }}$.

**Therefore, the correct answer is (D) ${{15}^{\circ }}$.**

**Note:**

Another method of solving this problem was by first finding the dot product of the two vectors of velocity of the billiard balls and then using that dot product to find the angle between them. For this method, we must have prior knowledge of finding the magnitude and dot product of vectors.

Recently Updated Pages

Which of the following materials is likely to have class 8 physics CBSE

What is a quasar What is their importance class 8 physics CBSE

Identify the parts which vibrate to produce sound in class 8 physics CBSE

Write the relation between liter and cm3 class 8 physics CBSE

Define the terms a ray of light and a beam of ligh class 8 physics CBSE

Suppose you are in a car and it is raining heavily class 8 physics CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail