# Physics Course/Projectile Motion

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## 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.

## 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