Answer
384.9k+ 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.
Complete answer:
![seo images](https://www.vedantu.com/question-sets/7e75bfa6-23dd-447b-9205-7f5ac6259e787361938647094372558.png)
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
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)