Kinematic Equations (2-4)
 v
a =
t
x
v = 
t
• Let's use these definitions for v and a to
derive four very useful relationships
between (x, v, a, t)
• Take t0 = 0 and x0 = 0 (∆t = t,  x = x )
• Assume acceleration a is constant
(1) Determine final velocity (v) from initial
velocity (v0), acceleration (a), and time (t)
(2) Determine position (x) from initial and
final velocities and time
v
a = 
t
(3) Determine position if we know initial
velocity, acceleration, and time
x
v = 
t
(4) Find an equation that doesn't involve
time
1. v = v0 + at
1. v = v0 + a t
2. x = ½(v0 + v)t
2. x = ½(v0 + v) t
Kinematics Equations for Constant
Acceleration
Let x=0 at t=0 (x0 = 0, t0 = 0)
1. v = v0 + at
2. x = ½ (v0 + v) t
3. x = v0t + ½ at2
4. v2 = v02 + 2ax
Notes: -- only true for constant acceleration!
-- v0 = starting value
-- x, v = end values
Problem Solving Strategy
• Read the problem carefully
• Draw a sketch of the situation
• Coordinate system with + and – dir'n
• Write down your “knowns” and
“unknowns” among x, vo, v, a, t
• Decide which equation(s) you will use
• Carry out the calculation
• Check: is your answer reasonable?
Example
A space probe starts from rest and is
launched towards Pluto with a constant
acceleration of 1x10-3 m/s2. If Pluto is 6x1012
m from the Earth, how long will it take the
probe to reach it?
1. v = v0 + at
2. x = ½ (v0 + v) t
3. x = v0t + ½ at2
4. v2 = v02 + 2ax
Example
David is driving a steady 30 m/s when he
passes Tina. Accelerating from rest at
2.0 m/s2, Tina catches up and passes
David.
(A) How far has Tina gone when she
passes David?
1. v = v + at
0
2. x = ½ (v0 + v) t
(B) What is her speed as
she passes him?
3. x = v0t + ½ at2
4. v2 = v02 + 2ax
1. v = v0 + at
2. x = ½ (v0 + v) t
3. x = v0t + ½ at2
4. v2 = v02 + 2ax
Example
A sprinter accelerates at 2.5 m/s2 until
reaching her top speed of 15 m/s. She
then continues to run at top speed. How
long does it take her to run the 100-m
dash?
1. v = v0 + at
2. x = ½ (v0 + v) t
3. x = v0t + ½ at2
4. v2 = v02 + 2ax