Question

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

Answer 1

Particle is projected with sped $u$ at an angle $\theta$. At the highest pt, the vertical component of velocity is $0$
So, that the point, velocity $=u\cos \theta$
centripetal force $=m{u}^2{\cos }^2\left(\frac{\theta }{r}\right)$
at hieghest pt
$mg=\frac{m{v}^2}{r}\Rightarrow r=\frac{m{v}^2}{r}\Rightarrow r=\frac{u^2{\cos }^2\theta }{g}$