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 \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 tEdit

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 flightEdit

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 rangeEdit

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 heightEdit

The maximum height is given by

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

where h is the initial height

method 1 - completing the squareEdit

method 2 - by symmetryEdit

method 3 - by calculusEdit

Last modified on 31 October 2012, at 22:06