This document provides an overview of displacement, velocity, acceleration, and free fall. It defines displacement as the change in position from starting to ending point and distinguishes it from distance traveled. Velocity is defined as a quantity that measures how fast an object moves and includes both speed and direction. Acceleration is the rate of change of velocity over time. For objects in free fall, gravity provides a constant downward acceleration of about 9.81 m/s2. Kinematic equations relate displacement, velocity, acceleration, and time for objects experiencing constant acceleration.

1. Lesson 2-1
Displacement and Velocity

2. Displacement
 Lets say you travel from here to Pittsburgh
 Many different ways, by boat, by car, by
plane
 Different methods mean different amounts of
time

3. Displacement
 End point is always the same
 To describe the results of your motion you need to specify
 Distance from starting point
 Direction of travel
 Direction and distance mean
 Displacement is a vector
 Back to the Pittsburgh example
 Displacement is the same no matter what method of travel
or how many stops, starts, or detours

4. Displacement
 Displacement is the change in position
 SI unit is the meter
 Usually talk about displacement of objects
that move
 An object at rest has zero displacement
 No matter how much time passes, the object will not
move

5. Displacement
 Displacement is NOT equal to distance
traveled
 Think of “Something moved around, what is the
shortest distance it could have taken?”
 Nascar races have zero displacement
 In football
 Offense hopes for positive displacement
 Defense hopes for negative displacement

6. Reference Points
 Coordinate systems are useful to describe
motion
 Yard markers help on a football field
 Squares on a chess board
 A meter stick is helpful to determine
displacement

7. Reference Points
 Lets say we have a ball
 The ball begins at 15 cm
 We refer to the starting point as xi
 The ball rolls to the 45 cm mark
 We refer to the ending point as xf
 Displacement is found by subtraction
 Final position – starting position

8. The Displacement Equation
 Final position –
starting position
 Recall Δ means
‘change in’
 The displacement
equation is:
x x xf i 

9. Direction of Displacement
 Displacement may also occur in the vertical
direction
 A helicopter sits on a heli-pad 30 m above the
ground, it takes off and hovers 200 m above the
ground
 What is the yi? What is the yf? What is the Δy?
 yi = 30 m, yf = 200 m, Δy = 170 m

10. Sign on Displacement
 Displacement may be positive or negative
 From our equation Δx = xf – xi we see
 Δx is positive if xf > xi
 Δx is negative if xf < xi
 There is no such thing as a negative distance
 A –Δx simply tells a direction

11. Sign on Displacement
 Coordinate directions
 Using ‘right’ as positive and ‘left’ as negative is
only by convention
 That does not mean it is necessarily correct
 As long as you remain constant throughout the
situation, you may call ‘left’ positive.
 Thus making ‘right’ negative
 Similarly, you may call ‘down’ positive
 Thus making ‘up’ negative

12. Displacement Practice
1) xi = 10 cm, xf = 80 cm
2) xi = 3 cm, xf = 12 cm
3) xi = 80 cm, xf = 20 cm
4) xi = 28 cm, xf = 11 cm
70 cm
9 cm
-60 cm
-17 cm
Concept Chall. Pg 41

13. Velocity
 Quantity that measures how fast something
moves from one point to another
 Different than speed, Velocity has direction
 Speed is the magnitude part of the velocity vector
 Velocity has direction and magnitude

14. Average Velocity
 To calculate, you must know the time the
object left and arrived
 Time from initial position to final position
 Avg. Vel. is displacement divided by total time
v
x
t
x x
t t
avg
f i
f i
 





15. Avg. Velocity vs Avg. Speed
 Main difference
 Average Velocity depends on total displacement
(direction)
 Average speed depends on distance traveled in a
specific time interval

16. Lesson 2-2
Acceleration

17. Acceleration
 Lets say you are driving at 10 m/s
 You approach a stop sign and brake carefully and
stop after 6 seconds
 Your speed changed from 10 m/s to 0 m/s over
that time
 Lets say you had to brake suddenly and stopped
after 2 seconds
 Your speed changed from 10 m/s to 0 m/s over
that time

18. Acceleration
 What was the main difference between those
two examples?
 Time
 A slow, gradual stop is much more comfortable than a
sudden stop

19. Average Acceleration
 The quantity that describes the rate of change
of velocity in a given time interval is
acceleration
a
v
t
v v
t t
avg
f i
f i
 





20. Average Acceleration
 Units of acceleration are length per seconds squared
 Analysis:
a
v
t
m s
s
m s s m savg     


/
/ / 2

21. Constant Acceleration
 As an object moves with constant a, the V
increases by the same amount each interval
 There is a very specific relationship between
displacement, acceleration, velocity, and time
 The relationship is used to produce a group of
very important equations

22. Kinematic Equation #1
 Displacement depends
on acceleration, initial
velocity and time and
v
x
t
avg 


v
v v
avg
f i


2
 


x
t
v vf i
2
 

 x
v v
t
f i
2
 x v v tf i 
1
2
( )Kinematic Equation #1:

23. Kinematic Equation #2
 Final velocity depends
on initial velocity,
acceleration and time
a
v
t
v v
t
f i
 

 
  a t v vf i
  v a t vi f
v v a tf i  Kinematic Equation #2:

24. Kinematic Equation #3
 We can form another
equation by plugging
#2 into #1
 x v v ti f 
1
2
( ) v v a tf i  
     x v v a t ti i
1
2
( ( ))
    x v a t ti
1
2
2( )
    x v a t ti( )
1
2
Kinematic Equation #3:
  x v t a ti ( )
1
2
2

25. Kinematic Equation #4
 So far, all of our Kinematic Equations have
required time interval
 What if we do not know the time interval
 We can form one last equation by plugging
equation #1 into #2

26. Kinematic Equation #4
 x v v ti f 
1
2
( )
  
L
NM O
QP2 2
1
2
 x v v ti f( )
  2 x v v ti f( )



2

x
v v
t
i f( )
v v a tf i  
  

L
N
MM
O
Q
PPv v a
x
v v
f i
i f
2
( )
  

L
N
MM
O
Q
PPv v a
x
v v
f i
i f
2
( )
   ( )( )v v v v a xf i i f 2 

27. Kinematic Equation #4
   ( )( )v v v v a xf i i f 2 
  v v a xf i
2 2
2 
Kinematic Equation #4: v v a xf i
2 2
2  
Note: A square root is needed to find the final velocity

28. Lesson 2-3
Falling Objects

29. Free Fall
 In a vacuum, with no air, objects will fall at
the same rate
 Objects will cover the same displacement in the
same amount of time
 Regardless of mass
 We cannot demonstrate this because of air
resistance

30. Gravity
 Objects in free fall are affected by what?
 Gravity
 A falling ball moves because of gravity
 “The force of gravity”
 Gravity is NOT a force!!
 Gravity is an acceleration

31. Gravity as an Acceleration
 Since acceleration is a vector
 Gravity has magnitude and direction
 Magnitude is -9.81 m/s2 or 32 ft/s2
 Direction is toward the center of the Earth
 Usually straight down
 Gravity is denoted as g rather than a
 Gravity is a special type of acceleration
 Always directed down, so the sign should always be
negative
 -9.81 m/s2 or -32 ft/s2

32. Path of Free Fall
 If a ball is thrown up in the air and falls back
down the same path, some interesting things
happen
 At the maximum height, the ball stops
 As the ball changes direction, it may seem as V and a
are changing
 V is constantly changing, a is constant from the
beginning
 a is g throughout

33. Path of Free Fall
 At ymax
 What is the velocity?
 0 m/s
 What is the acceleration?
 g or -9.81 m/s2

34. Free Fall
 It may be tough to think of something moving
upward and having a downward acceleration
 Think of a car stopping at a stop sign
 When an object is thrown in the air, it has a
+Vi and –a
 Since the two vectors are opposite each other, the
object is slowing down

35. Free Fall
 The velocity decrease until the ball stops and
velocity is 0
 It is tough to see the ‘stop’ since it is only for a
split second
 Even during the stop, a = -9.81 m/s2
 What happens after the ball stops at the top of
its path?

36. Free Fall
 The ball begins to free fall
 When the ball begins to move downward
 It has a negative velocity
 It has a negative acceleration
 V and a now in the same direction
 Ball is speeding up
 This is what happens to objects in free fall
 They fall faster and faster as they head toward
Earth