Question

A particle is projected with a speed u at an angle $\theta$ with the horizontal. Consider a small part of its path near the highest position and take it approximately to be a circular arc. What is the radius of this circle ? This radius is called the radius of curvature of the curve at the point.

Answer 1

Particle is projected with speed 'u' at an angle $\theta .$ At the highest point the vertical component of velocity is '0'.

So at the point, velocity = $ucos\theta$

centripetal force = $\frac{m{u}^{2}co{s}^2\theta }{r}$

At highest point

$mg=\frac{m{v}^2}{r}=\frac{m{u}^{2}co{s}^2\theta }{r}$

$\Rightarrow r=\frac{u^{2}co{s}^2\theta }{g}$