A projectile is projected from the origin with initial velocity
Equation of the trajectory is given by the kinematic expression
In the vertical direction.
Noting that the gravity is acting against the direction of motion
#y = (u sin theta)t - 1/ 2 g t^2#......(1)
In the horizontal direction.
Ignoring air resistance
#x = (u cos θ)t#........... (2)
To combine both equations.
Writing second term on the RHS in terms of
#y= x tan θ − (g x ^2)/( 2 v ^2 )( 1 + tan ^2 θ )#
Is the required equation of trajectory.