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:

$x^{\prime }(t)=v\cos \theta$

$y^{\prime }(t)=v\sin \theta$

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

$x(t)=v(\cos \theta )t$

$y(t)=v(\sin \theta )t-\frac{1}{2}g{t}^2$

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

$|\overrightarrow{v}|=\sqrt{[x^{\prime }(t){]}^2+[y^{\prime }(t){]}^2}$

and the direction is given by

$\tan \theta =\frac{x^{\prime }(t)}{y^{\prime }(t)}$

Time of flight

To solve for the time of flight, we set y(t)=0

$v(\sin \theta )t-\frac{1}{2}g{t}^2=0$

$t=0$ 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

$x(\frac{2v\sin \theta }{g})=\frac{v^2\sin 2\theta }{g}$

Maximum height

The maximum height is given by

$H=\frac{v^2{\sin }^2\theta }{2g}+h$

where h is the initial height