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

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

Maximum height

The maximum height is given by

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

where h is the initial height