A projectile is projected horizontally with velocity u and allowed to fall only under gravity. Discuss the trajectory of projectile. 

Consider a projectile, which is projected with velocity 'u' in horizontal direction from a point at height h from the ground. 
Let origin be the point of projection. 
The right direction is positive direction of X-axis and vertically downward direction is positive direction of Y-axis.
The component of velocity and acceleration is resolved into horizontal and vertical components.
So, 
                     
Motion along horizontal direction is, 
                   ...(1) 
Motion along vertical direction is,
                     ...(2)
Eliminating 't' from (1) and (2), we get
                         
This is the required equation of trajectory.
Since the equation of trajectory is an equation of parabola, therefore path of projectile is parabola.