A ball is projected with speed $u$ making an angle $ heta$ with the vertical. Consider a small part

Question

A ball is projected with speed $u$ making an angle $	heta$ with the vertical. Consider a small part of the trajectory 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).

Professor PC Thomas Classes · Answer

Angle with horizontal is $90^\circ - 	heta$. Velocity at highest point is purely horizontal $v = u\cos(90^\circ - 	heta) = u\sin	heta$. Normal acceleration is $g$. $R = v^2/a_n = \frac{u^2\sin^2	heta}{g}$.