Physics Course/Projectile Motion

Projectile Motion
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. all are due to gravity

Analysis (two dimensional space)
Suppose the object is projected at an angle $$\theta$$ 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:


 * $$ x'(t)=v \cos \theta $$
 * $$ y'(t)=v \sin\theta $$

By using $$s=vt+\frac{1}{2}at^2$$, The x- and y- coordinates of the object are


 * $$x(t)=v (\cos \theta)t$$
 * $$y(t)=v (\sin \theta)t-\frac{1}{2}gt^2$$

which are functions in time.

By eliminating t,
 * $$y(t)=(\tan \theta)x(t)-\frac{g}{2v (\cos \theta)}[x(t)]^2 +h$$

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
 * $$|\vec v|=\sqrt{[x'(t)]^2+[y'(t)]^2}$$

and the direction is given by
 * $$\tan \theta=\frac{x'(t)}{y'(t)}$$

Time of flight
To solve for the time of flight, we set y(t)=0
 * $$v (\sin \theta)t-\frac{1}{2}gt^2=0$$
 * $$t=0$$ or $$t=\frac{2v\sin\theta}{g}$$

Horizontal range
After $$t=\frac{2v\sin\theta}{g}$$, the x-coordinate of the object is given by
 * $$x(\frac{2v\sin\theta}{g})=\frac{v^2\sin2\theta}{g}$$

Maximum height
The maximum height is given by
 * $$H=\frac{v^2\sin^2\theta}{2g}+h$$

where h is the initial height