The equation of motion of a projectile is y=12x−34x2 The horizontal component of velocity is 3 ms-1. What is the range of the projectile?

Given y=12x−34x2,ux=3ms−1 vy=dydt=12dxdt−32xdxdt  At x=0,vy=uy=12dxdt−12ux=12×3=36ms−1 ay=ddtdydt=12d2xdt2−32dxdt2+xd2xdt2  But d2xdt2=ax=0. Hence  ay=−32dxdt2=−32u2x=−32×(3)2=−272ms−2  Range, R=2uxuyay=2×3×3627/2=16m  Alternatively: We have y=12x−34x2.   When projectile again comes to ground, y=0 and x=R.0=12R−34R2⇒R=16m