Bellaire High School Advanced Physics Chapter 2 - One Dimensional Motion

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