PROJECTILE MOTION, EVERYTHING YOU NEED TO KNOW

Projectile motion is a special case of two- dimensional constant acceleration.  It is special because the horizontal or x-component of acceleration can be approximated as zero.  The vertical or y-component of acceleration is approximated as. 

The initial x-position is defined as zero (x = 0).   The direction or the trajectory angle  is defined with respect to the horizontal.   is positive if the object is rising, it is zero if it is moving horizontally, and it is negative if it is falling. 
For > 0, we define the positive y-axis as up:   y and a =  = -g 
If   0, then y and a =  = + g. 	
v= v cos   and v = v sin  where v is the initial speed and  is the initial projection angle.

(1)	x = v t				Horizontal position as a function of time
(2)	y = y + vt + 1/2t		Vertical position as a function of time
(3)	v = v + t				Vertical velocity as a function of time
(4)	v=  v + 2		Scalar or time independent equation
(5)	y = 1/2 ( v+ v)t			Average vertical velocity and displacement
(6)	y = y + xtan +/(2vcos)		Path formula  y(x)
(7)	y = y + xtan + x[ 1 +tan]/(2v)	Double launch angle formula
(8)	t = v/g				Time to rise to maximum height
(9)	t = (2y/g)			Time down from max height to ground
(10)	h = y - y = v/2g		Height formula
(11)	x= R = v(t + t) 		Range formula  
(12)	R = vsin(2)/g			Range formula restricted for y = y 
(13)	tan = v/v			Trajectory angle formula
(14)	v = [(v)+ 2]		Final or landing speed
(15)	 = cos[v/v]			Final or landing angle

Equation (1) – (3) are fundamental.  

(4) is derived by solving for t in (3) and inserting t into (2). 

(5) is the result of a constant vertical acceleration.  

(6) is generated by solving  for t in (1) and inserting into (2);  x(t) and y(t), so y(x).  

(7) is derived from (6) using the trigonometry identity 1/cos = 1 + tan. 

(8) is derived from (3) and that v = 0 at y. 
(9) is derived from the free fall equation because at y,  v = 0. 

(10) is derived from (4) and v = 0 at y. 

(11) is based on (1) with the total time t = t + t.  

(12)  is derived using (8) – (11):  (8) = (9) for level flight and sin (2) = 2sincos.  

(13) follows from the definition of.

(14) is derived from (4) and v = v.

(15) Follows from the definition of cosine.