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  4. A particle is projected from the ground with an initial speed of v at an angle θ with horizontal. The average velocity of the particle between its point of projection and highest point of trajectory is

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

AIIMSAIIMS 2013Motion in a Plane

Solution:

From figure,
image
average velocity,
$v_{ av }=\frac{\sqrt{H^{2}+R^{2} / 4}}{T / 2}$...(i)
Here, $H=\frac{u^{2} \sin ^{2} \theta}{2 g}$
$R=\frac{u^{2} \sin 2 \theta}{g}$ and $T=\frac{2 u \sin \theta}{g}$
Putting these value in (i), we get
$v_{ av }=\frac{v}{2} \sqrt{1+3 \cos ^{2} \theta}$