Question

A particle is projected with a velocity u at an angle with the horizontal plane. Find the radius of curvature of the parabola traced out by the particle at the point where velocity makes an angle $\frac{\theta }{2}$ with the horizontal plane.

Answer 1

Given that the initial velocity of the particle is U. The object is projected at an angle θ. Then the horizontal and vertical components of the initial velocity are,

$Ux=Ucos\theta$

&

$Uy=Usin\theta$

Let after t time of projection, the particle is making an angle $\frac{\theta }{2}$. Therefore,

$Vcos\left(\frac{\theta }{2}\right)$ $=Ucos\theta$⇒V$=Ucos\theta cos\left(\frac{\theta }{2}\right)$

$=Vsin\left(\frac{\theta }{2}\right)$

$=\left(Ucos\theta cos\right(\frac{\theta }{2}\left)\right)\times sin\left(\frac{\theta }{2}\right)$