Download notes on Kinematics Equations

Uniform Accelerated Motion
Kinematic Equations
Kinematic Equations
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Kinematic Equations are considered to be
“equations of motion” and are based on the
fundamental definitions of average velocity and
acceleration:
𝑠
𝑣=
𝑡
v  v0
v
2
v  v0
a
t
Our variables
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There are 5 basic variables that are used in any
motion-related calculation:
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Initial Velocity = v0 or vi or v1 or u
Final Velocity = v or vf or v2
Acceleration = a
Displacement = d (sometimes also s or could be Dx)
Time = t
Bold face indicates a vector
Each of the kinematic equations will use 4 of
these 5 variables
Deriving the Equations
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Each of the kinematic equations starts with a
rearranged version of the equation for average
velocity:
s  v t
And uses substitution, rearranging, and
simplifying the equations to get to the end
result.
For example…
Kinematics Equation #1
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Step 1:
Step 2: Substitute
equation for v
Step 3: Rearrange
acceleration equation
to solve for t, then
substitute
Step 4: Simplify by
multiplying fractions
Step 5: Rearrange
s  v t
vu
v
2
t
v u
a
vu 
d


t
→
 2 
v u   v u 


 2   a 
→ s  
 v2  u 2 

s  
2
a


2as  v 2  u 2
v  u  2as
2
2
Kinematics Equation #2
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Step 1:
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Step 2: Substitute
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Step 3: Rearrange
acceleration equation to
solve for v, then
substitute
Step 4: Simplify
Step 5: Distribute the t
through the equation
Step 6: Simplify again
s  v t
vu
v
2
→
vu 
s
t
 2 
(u  at )  u 
v  u  at → s  
t
2


 2u  at 
s
t
 2 
 2ut  at 2 

s  
2


1 2
s  ut  at
2
Summary of Equations
v  u  at
v  u  2as
2
2
1 2
s  ut  at
2
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You will NOT be required to memorize these 
Lab Connection
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How might these equations relate to the
lab that we just completed?
How might you determine the acceleration
of your cart using the data collected?
What does your graph look like?
How might we change the axes in order to
have a linear fit?
Problem Solving Strategy
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When given problems to solve, you will be
expected to “show your work” COMPLETELY!
“Showing work” means that you will be expected to
include the following pieces in your full answer (or
you will not receive full credit for the problem…)
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List of variables – include units on this list
Equation – in variable form (no numbers plugged in yet)
If necessary, show algebra mid-steps (still no numbers)
Plug in your value(s) for the variables
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?
v = 0 m/s
v  u  at
u = 25 m/s
v  u  at
t = 3.0 s
v

u
a=?
a
t
0 m  25 m
s
s
a
3.0s
a = -8.3 m/s2
Practice Problem #2
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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?
v=
u=
t=
a=
s=
?
0 m/s
30.0 s
3.00 m/s2
?
1 2
s  ut  at
2
1
s  0  (3.00)(30.0) 2
2
s = 1350 m
v  u  at
v  0  (3.00)(30.0)
v = 90.0 m/s
Think Usain Bolt is fast?
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World’s fastest Mammal
Slow Motion Running