KINEMATIC EQUATIONS
New equations and how to use them!
DEFINITIONS
 Kinematics
– Effect of Motion
Study and description of motion –
without regard to the cause.
 Dynamics
– Cause of Motion
KINEMATIC EQUATIONS
 Equations
of motion
 Based on the fundamental definitions of
average velocity and average acceleration:
x
v
t
Arithmetic Mean
vi  vf
v
2
v
a
t
Plugging in def of Δ
(vf  vi )
a
t
OUR VARIABLES
 There
are 5 basic variables that are used in
any motion-related calculation:





Initial Velocity = v0 or vi
Final Velocity = v or vf
Acceleration = a
Displacement = Δx
Time = t
 Bold
face indicates a vector
 Each of the kinematic equations will use 4 of
these 5 variables
WHAT CAN WE DETERMINE?
 How
far does an object travel during uniformly
accelerated motion?
Start
x
v
t
Rearrange…
x  v t
Substitute…
 vf  vi 
x  
t
 2 
WHAT CAN WE DETERMINE?
Continue
 vf  vi 
x  
t
 2 
Rearrange…
(vf  vi )
a
t
Substitute…
 (a t  vi )  vi 
x  
t
2


Distribute Δt…
vf  a t  vi
To…

a t v t  v t 
x 
2
i
i
2
Combine like terms…
x  vit  a t
1
2
2
WHAT CAN WE DETERMINE?

Can we relate v, a, & Δx without a time variable?
Start
(vf  vi )
a
t
(vf  vi )
t 
a
Rearrange…
Substitute into…
To get..
x  v t
 (vf  vi ) 
x  v 

 a 
WHAT CAN WE DETERMINE?
Start
Substitute…
 (vf  vi ) 
x  v 

 a 
vi  vf
v
2
 (vf  vi )  (vf  vi ) 
x  


 2  a 
To get..
Multiply binomials…
Solve for vf…
2
2
 (vf  vi ) 
x  

 2a

vf 2  vi 2  2a x
SUMMARY OF EQUATIONS
(vf  vi )
a
t
No
Position
x  v t
No Time
 You
x  vit  a t
1
2
vf
2

2
vi  2a x
2
will NOT be required to memorize these 
LAB CONNECTION: BUGGY LAB
 The
equation of the position vs. time graph is:
xf  vt  xi
The slope of this graph = velocity
The y-intercept of this graph = initial position
LAB CONNECTION: GIP’ER LAB
 The
equation of the velocity-time graph is:
vf  at  vi
The slope of this graph = acceleration
The y-intercept of this graph = initial velocity
EQUATIONS THAT DESCRIBE OBJECTS THAT
CHANGE THEIR VELOCITIES:
Linear
Graphs from Lab Equations from data
X vs. t2
V vs. t
V2
vs. X
cm s  2

x   0.3
t
s 

cm

V   0.6
s

s
 t  Vo

 cm
V   1 .2
s

2
s
x

General Equation
1 2
x  at
2
V  at  Vo
V 2  2ax
PROBLEM SOLVING STRATEGY
 Show
your work – ALWAYS!
 Sketch
 Use


three step method:
Equation in variable form (no numbers plugged in yet)


of situation, motion map, x vs. t plot
If necessary, show algebra mid-steps (still no numbers)
Equation with value(s) for the variables (numbers!)
Final answer: boxed/circled with appropriate units
and sig figs
PRACTICE PROBLEM #1
A
school bus is moving at 25 m/s when the
driver steps on the brakes and brings the bus
to a stop in 3.0 s. What is the average
acceleration of the bus while braking?
vf = 0 m/s
vf  vi  at
vi = 25 m/s
vf  vi  at
Δt = 3.0 s
v
f  vi
a= ?
a
t
0 m  25 m
s
s
a
3.0s
a = -8.3 m/s2
PRACTICE PROBLEM #2
 An
airplane starts from rest and accelerates at
a constant 3.00 m/s2 for 30.0 s before leaving
the ground.
(a) How far did it move?
(b) How fast was it going when it took off?
vf = ?
vi = 0 m/s
Δt = 30.0 s
a = 3.00 m/s2
Δx = ?
1 2
x  vi t  at
2
1
x  0  (3.00)(30.0) 2
2
Δx= 1350 m
vf  vi  at
v  0  (3.00)(30.0)
v = 90.0 m/s