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Question

(C) The average velocity is defined as the total displacement divided by the total time taken.Let the initial position be $(0, 0)$. The highest point

Vedclass · Accepted Answer

$\frac{v}{2} \sqrt{1+3 \cos ^2 	heta}$

Vedclass · Answer

(C) The average velocity is defined as the total displacement divided by the total time taken.Let the initial position be $(0, 0)$. The highest point is reached at time $t = \frac{v \sin 	heta}{g}$.At this time,the horizontal displacement is $x = (v \cos 	heta) t = \frac{v^2 \sin 	heta \cos 	heta}{g} = \frac{R}{2}$,and the vertical displacement is $y = H = \frac{v^2 \sin^2 	heta}{2g}$.The displacement vector is $\vec{s} = \frac{R}{2} \hat{i} + H \hat{j}$.The magnitude of displacement is $|\vec{s}| = \sqrt{(\frac{R}{2})^2 + H^2}$.Substituting $R = \frac{v^2 \sin 2	heta}{g} = \frac{2v^2 \sin 	heta \cos 	heta}{g}$ and $H = \frac{v^2 \sin^2 	heta}{2g}$:$|\vec{s}| = \sqrt{(\frac{v^2 \sin 	heta \cos 	heta}{g})^2 + (\frac{v^2 \sin^2 	heta}{2g})^2} = \frac{v^2 \sin 	heta}{g} \sqrt{\cos^2 	heta + \frac{\sin^2 	heta}{4}} = \frac{v^2 \sin 	heta}{2g} \sqrt{4 \cos^2 	heta + \sin^2 	heta} = \frac{v^2 \sin 	heta}{2g} \sqrt{3 \cos^2 	heta + 1}$.The time taken is $t = \frac{v \sin 	heta}{g}$.Therefore,average velocity $v_{av} = \frac{|\vec{s}|}{t} = \frac{\frac{v^2 \sin 	heta}{2g} \sqrt{3 \cos^2 	heta + 1}}{\frac{v \sin 	heta}{g}} = \frac{v}{2} \sqrt{1 + 3 \cos^2 	heta}$.

$A$ particle is projected from the ground with an initial speed of $v$ at an angle of projection $\theta$. The average velocity of the particle between its time of projection and the time it reaches the highest point of its trajectory is

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