Motion In A Plane

§§ Radius of curvature at any point on the path of a projectile
          
 

Let at any time t the velocity vector \(\vec v\) be inclined at an angle \(\theta\) with horizontal at a point say, P as shown in figure. Since gravitational acceleration g is always acting vertically downwards the component of g perpendicular  to the velocity vector \(\vec v\) can be treated as a  radial acceleration towards, the centre of a circular path of radius (let r) as shown in figure. r is known as radius of curvature of the parabola at p 
    \( \overrightarrow v \bot \overrightarrow a \,\,and\,a_r = \frac{{v^2 }} {r},when\,\,a_r = g\cos \theta \)  
   \( \because \,\,a_r = \frac{{v^2 }} {r}\,\,\,\, \Rightarrow r = \frac{{v^2 }} {{a_r }},\,\,where\,\,a_r = g\cos \theta \, \)
        \( \Rightarrow r = \frac{{v^2 }} {{g\cos \theta }} \)