Q. Two stones are projected from the top of a tower in opposite direction, with the same velocity v but at 30° & 60° with horizontal respectively. The relative velocity of first stone relative to second stone is

(a) 2v

(b) √2 v

(c) 2 v/√3

(d) v/√2

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Sol: $\displaystyle \vec{v_1} = v cos30\hat{i} + v sin30\hat{j}$

$ \displaystyle \vec{v_2} = v cos60\hat{(-i)} + v sin60\hat{j}$

$ \displaystyle \vec{v_{12}} = \vec{v_1}-\vec{v_2} $

$ \displaystyle = (v cos30 + v cos60)\hat{i} + (v sin30 – v sin60)\hat{j}$

$ \displaystyle = v[(\frac{\sqrt{3}}{2} +\frac{1}{2})\hat{i} + (\frac{1}{2} -\frac{\sqrt{3}}{2})\hat{j}] $

$ \displaystyle = v[ \frac{\sqrt{3}+1}{2}\hat{i} – \frac{\sqrt{3}-1}{2}\hat{j}]$

$ \displaystyle v_{12} = \sqrt{2} v $