Equations for Ch.3 Kinematics in 2-D
For X components ax = 0
For Y components ay = g =
vix = vfx = vx
dx = vx . t
vfy = viy + g.t
dy = (viy + vfy).t / 2
---------------------
dy = viy.t + ½ g.t2
vx = V cos 
vfy2 = viy2 + 2.g.dy
-9.8 m/s2
vY = V sin 
= tan-1 (vy / vx)
v=
vx 2  vy 2
Use the absolute value of g as 9.8 m/s2, in 4 equations below; That is, time
to apex, time of flight, maximum height and range of a projectile.
Time to the apex
ta 
vi sin 
g
Time of flight (hang time)
tf 
2vi sin 
g
Maximum height of a projectile
Range of a projectile
v 2 sin 2   vi sin  
h i

2g
2g
2
vi2 sin 2
R
g
For the two equations below, use g = -9.8 m/s2. The below two equations
dy and velocity = vfy, and are valid only
if initial vertical velocity is zero, viy = 0
2 dY
t
Time for a particle to fall height dy
g
Velocity of a particle after falling height dy
vfy  2 gdY = gt
are only for vertical components.
Ch 3 Handouts Kinetmatics in 2-D
Projectile motion is kinematics in 2-D, when the object only experiences acceleration (aY
= -9.8 m/sec2) due to gravity. Neglecting air resistance, aX = 0, the horizontal component
of the velocity vX would never change. Only vY is affected by gravity and continuously
decreases on the way up reaching zero Y-velocity at the top, and then increasing in the
downward direction. However remember that X-velocity is still the same, so the resultant
velocity is the Pythagorean sum of the X & Y velocities. The magnitudes of the Yvelocities are the same at equal heights and opposite in directions. Time only depends
on the Y-components of motion. The motion of a projectile can be analyzed by treating
the X and Y components of the vector separately and realizing that time ‘t’ is the same
for both. The equations of kinematics are written only for the Y-components. There is
only one equation for X-components as there is no acceleration for X-direction.
If a projectile is thrown with initial velocity vi and angle then
vX = vi . cos 
viY = vi . sin 

At the top vFY = 0, vX does not change during the entire flight.
Remember that X & Y components of motion are completely independent of each other,
and time is the only variable that can be shared between the two.
The time to apex, time of flight, Range and maximum height equations are only valid for
a full range. Below are examples where those equations cannot be used.