Useful Equations for Projectile Motion

for any occasion

to use an equation

to figure how far a ball goes.

or maybe how high,

and the angle it flies

or velocity, now you will know.

so if we can take
an example and say,
when you multiply one half of g
with the square of the time,
the position of y
is the answer invariably

how easy it is
when the formulas give
the projectile motion you need.
just plug in the numbers,
there'll be no more blunders
in solving the problems you see.

^{RolandMoutal designed this using GeoGebra, I highly recommend his practical and informative creation for anyone who is interested in projectile motion. You can find it here.}

Terms defined:

Vo = original velocity

α = angle

g = gravity (9.8m/s)

x = position of x along x-axis

y = position of y along y-axis

t = time

R = range max. or distance of x

h = height max. of y

Equations:

to find the velocity of x when original velocity and angle are known:

Vox = Vo cos α

to find the velocity of y when original velocity and angle are known:

Voy = Vo sin α

to find the angle when the velocity of x and  y are known:

 

   

to find the angle when only the range and velocity are known: 

 

  

up to 45°, after that the equation becomes:

 

  

to find the angle when only height and velocity are known:

 

to find the position of x at a certain time when the velocity of x is known:

x = Voxt

to find the position of y at a certain time when the original velocity of y is known:

 

the velocity of x is found in the above equation (Vo cos α):

Vx =  Vox

to find the velocity of y at a certain time when the original velocity of y is known:

Vy = Voy – gt

max height for y can be found by:

 

then:

 

to find velocity when the velocity of x and y are known:

(the pathagorean theorem a² + b² = c²)

to find the velocity when only the range and angle are known:

to find the velocity when only height and angle are known:

to find the time when y will be at a certain position (Δy) when that certain position and y’s original velocity are known: