Physics Course/Projectile Motion

Projectile MotionEdit

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)Edit

Suppose the object is projected at an angle ${\displaystyle \theta }$  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:

${\displaystyle x'(t)=v\cos \theta }$
${\displaystyle y'(t)=v\sin \theta }$

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

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

which are functions in time.

By eliminating t,

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

which shows that the trajectory is a parabola

Velocity at any time tEdit

The magnitude of the velocity at any time t is given by

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

and the direction is given by

${\displaystyle \tan \theta ={\frac {x'(t)}{y'(t)}}}$

Time of flightEdit

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

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

Horizontal rangeEdit

After ${\displaystyle t={\frac {2v\sin \theta }{g}}}$ , the x-coordinate of the object is given by

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

Maximum heightEdit

The maximum height is given by

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

where h is the initial height