Required, →v.→u=0
or, (ux^i+(uy−gt)^j).(ux^i+uy^j)=0
which gives, u2x+u2y−uygt=0
or, u2cos2θ+u2sin2θ−ugt.sinθ=0
Thus, t=ugsinθ
Now, trajectory of projectile is given as:
x=ucosθ.t
y=usinθ.t−12gt2=xtanθ−12g.x2u2cos2θ
Thus we have,
R=[1+(dydx)2]3/2∣∣∣d2ydx2∣∣∣
=[1+(tanθ−gxu2cos2θ)2]3/2∣∣∣−gu2cos2θ∣∣∣
Now, at t=ugsinθ, x=ucosθ.ugsinθ=u2gcotθ
Put this in equation for R, we get:
R=[1+(tanθ−cotθcos2θ)2]3/2∣∣∣−gu2cos2θ∣∣∣