Question

Two stones are thrown simultaneously from the same point with equal speed ${{V}_{0}}$at an angle $30{}^\circ$ and $60{}^\circ$ respectively with the positive axis. The velocity with which the stones move relative to each other in air is.
$A)2{{v}_{0}}\cos 15{}^\circ$
$B)2{{v}_{0}}\sin 15{}^\circ$
$\text{C)2}{{\text{v}}_{\text{0}}}\text{tan15 }\\!\\!{}^\circ\\!\\!\text{ }$
$\text{D)2}{{\text{v}}_{\text{0}}}\text{cot15 }\\!\\!{}^\circ\\!\\!\text{ }$

Answer 1

Here, two stones are thrown at different angles. The stones are moving in the same direction. Then, the relative velocity is the difference of the velocities of stone 1 and 2. This resultant velocity can be calculated by finding the x component of both stones velocity, and y component of both stones velocity. And then using the x and y component of the resultant velocity, we can find its magnitude.

Formula used:
${{v}_{x}}=v\text{ cos}\theta$
${{v}_{y}}=v\text{ sin}\theta$
$\left| {\vec{v}} \right|=\sqrt{v_{x}^{2}+v_{y}^{2}}$
${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
$\cos \left( A+B \right)+\cos \left( A-B \right)=2\cos A\cos B$
$\cos \left( A-B \right)-\cos \left( A+B \right)=2\sin A\sin B$
$\left( 1-\cos \theta =2{{\sin }^{2}}\dfrac{\theta }{2} \right)$

Complete answer:
Given,
Stone 1 is thrown at an angle of$30{}^\circ$, with velocity${{v}_{0}}$. Then,
X-axis component of the velocity is,${{\left( {{v}_{x}} \right)}_{1}}={{v}_{0}}\cos \theta ={{v}_{0}}\cos 30{}^\circ$
Y-axis component of the velocity is,${{\left( {{v}_{y}} \right)}_{1}}={{v}_{0}}\sin \theta ={{v}_{0}}\sin 30{}^\circ$
Stone 2 is thrown at an angle of $60{}^\circ$, with velocity${{v}_{0}}$ Then,
X-axis component of the velocity is, ${{\left( {{v}_{x}} \right)}_{2}}={{v}_{0}}\cos \theta ={{v}_{0}}\cos 60{}^\circ$
Y-axis component of the velocity is, ${{\left( {{v}_{y}} \right)}_{2}}={{v}_{0}}\sin \theta ={{v}_{0}}\sin 60{}^\circ$
Since the two stones move in the same direction, the relative velocity is the difference of the two velocities.
Then,
Relative velocity of two stones with respect to x axis,
${{\left( \text{Relative velocity} \right)}_{x}}\text{ = }{{\text{v}}_{0}}\cos 60-{{\text{v}}_{0}}\cos 30={{\text{v}}_{0}}\left( \cos 60-\cos 30 \right)$
Relative velocity of two stones with respect to y axis,
${{\left( \text{Relative velocity} \right)}_{y}}\text{ = }{{\text{v}}_{0}}\sin 60-{{\text{v}}_{0}}\sin 30={{\text{v}}_{0}}\left( \sin 60-\sin 30 \right)$
Then,
Resultant relative velocity,
${{\left( Velocity \right)}_{\text{resultant}}}\text{= }{{\text{v}}_{0}}\left( \cos 60-\cos 30 \right)+{{\text{v}}_{0}}\left( \sin 60-\sin 30 \right)$
We have,
Magnitude of a vector $\vec{v}$ is given by,
$\left| {\vec{v}} \right|=\sqrt{v_{x}^{2}+v_{y}^{2}}$
Where,
${{v}_{x}}=v\text{ cos}\theta$, ${{v}_{y}}=v\text{ sin}\theta$ and, $\theta$is the angle between ${{\text{v}}_{\text{x}}}\text{ and v}$
Then,
$\left| {{\left( Velocity \right)}_{\text{resultant}}} \right|=\sqrt{{{v}_{0}}^{2}{{\left( \cos 60-\cos 30 \right)}^{2}}+{{v}_{0}}^{2}{{\left( \sin 60-\sin 30 \right)}^{2}}}$
$={{v}_{0}}\sqrt{\text{co}{{\text{s}}^{\text{2}}}\text{60+si}{{\text{n}}^{\text{2}}}\text{60+co}{{\text{s}}^{\text{2}}}\text{30+si}{{\text{n}}^{\text{2}}}\text{30-2cos60cos30-2sin60sin30}}$
We have,
${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$,
$\cos \left( A+B \right)+\cos \left( A-B \right)=2\cos A\cos B$
$\cos \left( A-B \right)-\cos \left( A+B \right)=2\sin A\sin B$
Then,
$\left| {{\left( Velocity \right)}_{\text{resultant}}} \right|={{v}_{0}}\sqrt{1+1-2\left( \cos \left( 60-30 \right) \right)}={{v}_{0}}\sqrt{2-2\cos 30}$
$={{v}_{0}}\sqrt{2\left( 1-\cos 30 \right)}$
$={{v}_{0}}\sqrt{2\times 2{{\sin }^{2}}15}$ $\left( 1-\cos \theta =2{{\sin }^{2}}\dfrac{\theta }{2} \right)$
$={{v}_{0}}\sqrt{4{{\sin }^{2}}15}$
$=2{{v}_{0}}\sin 15{}^\circ$
Therefore, the velocity with which the stones move relative to each other in air is $2{{v}_{0}}\sin 15{}^\circ$

Answer is option B.

Note:
The velocity of a body with respect to the velocity of another body is known as the relative velocity of the first body with respect to the second.
If ${{V}_{A}}$ and ${{V}_{B}}$ represent the velocities of two bodies A and B respectively, then the relative velocity of A with respect to B is represented by ${{V}_{AB}}$.Similarly, the relative velocity of B with respect to A is represented by ${{V}_{BA}}$.
If the two objects are moving in the same direction, the relative velocity is given by,
${{V}_{AB}}={{V}_{A}}-{{V}_{B}}$, ${{V}_{BA}}={{V}_{B}}-{{V}_{A}}$
And if the two objects are moving in the opposite direction, then their relative velocity is given by,
${{V}_{AB}}={{V}_{BA}}={{V}_{A}}+{{V}_{B}}$