Projectile Motion refers to any motion moving under the effect of gravity. This kind of motion is famous for its trajectory being in the shape of a parabola.
Analysis (two dimensional space)
Suppose the object is projected at an angle at a height h with an initial velocity of v with a gravity of g. When on Earth g will equal 9.8 m/s2.
The components of velocity in horizontal (x-) and vertical (y-) directions are:
By using , The x- and y- coordinates of the object are
which are functions in time.
By eliminating t,
which shows that the trajectory is a parabola
Velocity at any time t
The magnitude of the velocity at any time t is given by
and the direction is given by
Time of flight
To solve for the time of flight, we set y(t)=0
After , the x-coordinate of the object is given by
The maximum height is given by
where h is the initial height
method 1 - completing the square
method 2 - by symmetry
method 3 - by calculus