Four equations of motion

For constant acceleration, you have now learned two equations of motion: equations (4.2) and (4.3). These two equations can be used to solve a wide variety of motion problems in physics. In some problems, however, it might be more convenient to use two additional equations. These two additional equations aren’t unique or different from the other ones, they are just more convenient to use in certain circumstances. As you will see below, we get the two new equations by manipulating the two equations you already know!
Start with equation (4.3):
x = x 0 + v 0 t + 1 2 a t 2
Assuming that we are starting at an initial time t = 0, acceleration is the change in velocity divided by the change in time or a = (v − v0)/t. Use this equation to substitute for acceleration in equation (4.3) to get
x = x 0 + v 0 t+ 1 2 ( v v 0 t ) t 2 = x 0 + v 0 t+ 1 2 vt 1 2 v 0 t = x 0 + 1 2 ( v 0 +v )t
This third equation of motion is useful when you don’t know the acceleration.
(4.4) x= x 0 + 1 2 ( v 0 +v )t
x  = position (m)
x0  = initial position (m/s)
v0  = initial velocity (m/s)
v  = velocity (m/s)
t  = time (s)
Position
when acceleration
is not known
Show Understanding the third equation
For the fourth equation of motion, start with equation (4.4) above, but rewrite it as
x x 0 = 1 2 ( v 0 +v )t 2( x x 0 )=( v 0 +v )t
This time we want to remove the dependence on time. Rewrite the definition of acceleration to solve for time
a= v v 0 t t= v v 0 a
and substitute this equation for acceleration to get
2( x x 0 )=( v 0 +v )( v v 0 a ) 2( x x 0 )= v 2 v 0 2 a 2a( x x 0 )= v 2 v 0 2
Rearranging these terms leads to equation (4.5), which is useful for solving problems when you don’t know the time.
(4.5) v 2 = v 0 2 +2a( x x 0 )
v  = velocity (m/s)
v0  = initial velocity (m/s)
a  = acceleration (m/s2)
x  = position (m)
x0  = initial position (m)
Velocity
when time is
not known
Show Understanding the fourth equation
Equations (4.2), (4.3), (4.4), and (4.5) are the four equations of motion.

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