What do you understand by projectile ? Show that the path of a projectile is parabolic.

Explanation

Projectile: Projectile is defined as a body thrown with some initial velocity and then allowed to move under the action of gravity alone, without being propelled by an engine or fuel or any source.
Path of Projectile : Suppose a body is projected up with an initial velocity u in a direction making an angle`theta` with the horizontal. Let O be the point of projection of the body and OX Fand.oy the horizontal and vertical. Resolving u in horizontal and vertical components.
We have
`u _(x) = u cos theta`
`u _(y) = u sin theta `

The body moves under constant gravitational acceleration 8 acting vertically downwards. Therefore the horizontal velocity uy remains unchanged throughout the motion, but the vertical velocity u, continuously changes. The displacement of the body along the horizontal direction after a time t is:
x= Horizontal velocity `u _(x) xx` time
`x = u cos theta xx t " "...(i)`
The displacement along the vertical direction is :
`y = ( u sin theta) xx t - (1)/(2) g t ^(2) " "...(ii)`
putting `t = (u)/( u cos theta) ` in eqn (ii) we get
`y = u sin theta x (x)/( u cos theta ) - (1)/(2) g xx ((x )/( u cos theta )) ^(2)`
`y = x tan theta - (1)/(2) g xx (gx ^(2))/(u ^(2) cos ^(2) theta)`
It represents a parabola. Hence the path of the projectile is parabola.