First slide

Rectilinear Motion

Question

A body is projected from the ground with a velocity u at an angle $θ$ with the horizontal. The average velocity of the body between its point of projection and the highest point of its trajectory is

Difficult

A

$\frac{u}{2} (1 + \cos θ)$
B

$\frac{u}{2} {(1 + \cos^{2} θ)}^{1 / 2}$
C

$\frac{u}{2} {(1 + 2 \cos^{2} θ)}^{1 / 2}$
D

$\frac{u}{2} {(1 + 3 \cos^{2} θ)}^{1 / 2}$

Solution

A is the highest point on the trajectory.
$Average velocity = \frac{displacement OA}{time taken}$
$= \frac{\sqrt{h_{max}^{2} + \frac{R^{2}}{4}}}{\frac{1}{2} t_{f}} . . . . . . . . . (1)$
Where $h_{max} = \frac{u^{2} \sin^{2} θ}{2 g}$
$R = \frac{u^{2} \sin 2 θ}{g},$ and $t_{f} = \frac{2 u \sin θ}{g}$
Using these in Eq. (1) and simplifying, we get
Average velocity $= \frac{u}{2} {(1 + 3 \cos^{2} θ)}^{1 / 2}$

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