At some instant during projectile motion, the speed of the projectile is v and its velocity makes an angle φ with the horizontal. The radius of curvature at that instant is:
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Question

At some instant during projectile motion, the speed of the projectile is v and its velocity makes an angle φ with the horizontal. The radius of curvature at that instant is:

G1 Physics · Accepted Answer

Gravity has magnitude g vertically downward. The component of gravity perpendicular to the instantaneous velocity is aₙ = g cosφ. Radius of curvature is defined by aₙ = v²/R. Therefore R = v²/aₙ = v²/(g cosφ).