1. Chapter 2: Motion in One Dimension
Section 1: Displacement & Velocity

2. Mechanics Kinematics Dynamics
Describes the motion of objects without looking at the cause of the motion
Dynamics
Deals with the effect that forces have on motion

3. Motion
the change of an object’s position relative to some reference point
Objects that are “at rest” are not in motion…obviously 

4. Describing Motion Most basic information needed: position and time
Whether or not an object is in motion depends on the reference point you choose.

5. Measuring Motion Scalar quantity Vector
Have magnitude (numerical value) but no direction
Vector
Have magnitude and direction
“Quantity” is the root word for quantitative… you get a number answer

6. Distance Distance (d) – how far an object travels.
Does not depend on direction.
Imagine an ant crawling along a ruler.
What distance did the ant travel?
d = 3 cm cm 1 2 3 4 5 6 7 8 9 10

7. Distance Distance does not depend on direction.
Here’s our intrepid ant explorer again.
Now what distance did the ant travel?
d = 3 cm
Does his direction change the answer? cm 1 2 3 4 5 6 7 8 9 10

8. Distance Distance does not depend on direction.
Let’s follow the ant again.
What distance did the ant walk this time?
d = 7 cm cm 1 2 3 4 5 6 7 8 9 10

9. Displacement: ∆x = xf – xi
Displacement – change in position of an object – measures how far an object has moved from its starting location
Note: ∆ is the Greek symbol “delta”. It means “a change in”.
xf means final position.
xi means initial position.
Displacement: ∆x = xf – xi
Examples of directions:
+ and –
N, S, E, W
Angles
In order to define displacement,
we need a direction
(depends on your frame of reference)

10. Displacement vs. Distance
Example of distance:
The ant walked 3 cm.
Example of displacement:
The ant walked 3 cm EAST.
An object’s distance traveled and its displacement aren’t always the same!

11. Displacement
Let’s revisit our ant, and this time we’ll find his displacement.
Distance: 3 cm
Displacement: +3 cm
The positive gives the ant a direction! cm 1 2 3 4 5 6 7 8 9 10 + -

12. Displacement Find the ant’s displacement again. - + Distance: 3 cm
Remember, displacement has direction!
Distance: 3 cm
Displacement: -3 cm cm 1 2 3 4 5 6 7 8 9 10 + -

13. Displacement Find the distance and displacement of the ant. - +
Distance: 7 cm
Displacement: +3 cm cm 1 2 3 4 5 6 7 8 9 10 + -

14. Displacement vs. Distance
An athlete runs around a track that is 100 meters long three times, then stops.
What is the athlete’s distance and displacement?
Distance = 300 m
Displacement = 0 m
Why?

15. With your partner, discuss the following:
What is this object’s position at 1 minute?
What is the object’s position at 2 minutes?
What is its displacement between 1 min and 5 min?
Describe the motion of this object.

16. Speed vs Velocity

17. Speed Speed (s) – Rate at which an object is moving.
speed = distance / time
s = d/t
Like distance, speed does not depend on direction.

18. Speed A car drives 100 meters in 5 seconds.
What is the car’s average speed?
s = d/t
s = (100 m) / (5 s) = 20 m/s
100 m
5 s
4 s
1 s
2 s
3 s

19. Velocity: v = ∆x = (xf-xi)
Average velocity – the total displacement divided by the total amount of time during the displacement.
Velocity may be + or - , depending on the displacement.
Velocity: v = ∆x = (xf-xi)
∆t (tf – ti)

20. Why refer to it as average velocity?
Example:
Suppose you traveled from your house to school…a distance of 4.0 km. It took you .20 hours (12 minutes) because of heavy traffic.
Your avg velocity would be:
v = 4.0 km / .20 h
v = 20 km/h
Did you travel at that exact speed for the entire trip?
Of course not.
Because of this, velocity is an average.

21. Velocity and speed are not the same:
Velocity requires direction.
Speed refers to the numerical value of velocity.

22. Avg. Speed and Avg. Velocity
I walk 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and average velocity.
Average speed: 0.50m/s
Displacement is 0 meters, average velocity is 0m/s

23. Pulling It All Together
Back to our ant explorer!
Distance traveled: 7 cm
Displacement: +3 cm
Average speed: (7 cm) / (5 s) = 1.4 cm/s
Average velocity: (+3 cm) / (5 s) = +0.6 cm/s cm 1 2 3 4 5 6 7 8 9 10 + -
5 s
4 s
1 s
2 s
3 s

24. Example
A crow flies east in a straight line. It begins flying at 3:04s, and comes to a rest at 3:11s. It was flying at an average velocity of 1.5 m/s. How far did the crow move from its initial position (in meters)?

25. (Final) Displacement: xf = v∆t + xi
Using the velocity equation, we can derive another equation to find an object’s position
With your partner…
1) Use the base velocity equation to derive an equation to solve for the final displacement of an object.
(Final) Displacement: xf = v∆t + xi

26. The slope of the line corresponds to the velocity.
Velocity can be determined using a position vs time graph.
The slope of the line corresponds to the velocity.

27. Position vs Time How could we describe the motion of these objects?

28. Review Constant Velocity Positive Velocity
Positive Velocity Changing Velocity (acceleration)
Slow, Rightward(+) Constant Velocity
Fast, Rightward(+) Constant Velocity

29. x t A B C
Graphing !
1 – D Motion
A … Starts at home (origin) and goes forward slowly
B … Not moving (position remains constant as time progresses)
C … Turns around and goes in the other direction quickly, passing up home

30. What is the AVERAGE speed of the bass boat depicted in the graph?
Average speed is taking the total distance traveled (0 to 125 meters), and dividing by the total time (1 to 9 seconds) it takes.

31. What is the instantaneous speed of the bass boat at t=7 seconds?
Instantaneous Speed = 85 meters = 12.1 m/s
7 seconds

32. Example
Sketch a position vs. time and a velocity vs. time graph for the following:
Your mother walks five meters to the kitchen at a velocity of 2 m/s, pauses by the refrigerator for three seconds, then walks back with a plate full of potato salad at a velocity of 1 m/s.

33. At track practice, the coach makes the team members run back and forth between two lines three times

34. A car is driving down the road at 55 mph, and after getting a flat tire moves at 35 mph

35. A bullet is fired from a gun and travels 1000 meters in one second before hitting a solid target. Sketch a position vs time and a velocity vs time graph for this scenario.

36. When you hit a golf ball, it travels through the air at 25 m/s, then decelerates once it hits the ground 100 meters away until it comes to a stop 110 meters away

37. Chapter 2: Motion in One Dimension
Section 2: Accelerated Motion

38. Acceleration: a = ∆v/∆t = vf - vi
Definition: acceleration – the change in velocity over time.
An object accelerates when it changes its motion.
This means:
Speeding up
Slowing down
Changing direction
Acceleration: a = ∆v/∆t = vf - vi
tf - ti

39. Acceleration has the derived unit of m/s2.
But…what does this unit actually mean?
With your partner…
Use the acceleration formula to show how the unit of m/s2 is obtained.
Explain, in plain speak, what the unit m/s2 means.
m 1
s s x =
_m_
s x s s2 m
_s_ s

40. The sign tells the direction.
The magnitude of acceleration tells how quickly the change is happening.
The sign tells the direction.
Acceleration with a “+” magnitude means the object is gaining velocity in the “+” direction.
Acceleration with a “–” magnitude means the object is slowing down (or gaining velocity in the “–” direction.

41. Objects that have a constant velocity have no acceleration.
Centripetal acceleration – the constant change of direction of an object moving in circles.

42. Example
Turner’s treadmill runs with a velocity of -1.2m/s and speeds up at regular intervals during a half hour workout. After 25 min, the treadmill has a velocity of -6.5 m/s. What is the average acceleration of the treadmill during this period?

43. Example
A shuttle bus slows down with an average acceleration of -1.8m/s/s. How long does it take the bus to slow from 9.0m/s to a complete stop?

44. Acceleration can be determined using a velocity vs time graph.
Use the image below to:
Calculate the average velocity of the car.
Create a velocity vs time graph for the car.
Calculate the avg. acceleration of the car.

45. Velocity vs Time
The slope of a velocity vs time graph represents acceleration.

46. Displacement can be figured out by using a velocity vs time graph
The area bound by the line and axes represent displacement
Trapezoid:
A = 1/2b*(h1+h2)
Rectangle:
A = b*h
Triangle:
A = 1/2b*h

47. Calculate the Displacement
Area = 40m
Area = 180m
Area = 80m

48. Using the base equation for acceleration, we can find other equations to find the velocity and/or displacement of moving objects that are changing their speed…
With your partner:
Using the base equation for acceleration, derive two new
equations to solve for initial displacement and final velocity.

49. Bellringer: 1/22 What does the red line represent?
What is the acceleration at A? B? C?

50. Graphing w/ Accelerationx
Graphing w/ Acceleration C B t A D
A … Start from rest south of home; increase speed gradually
B … Pass home; gradually slow to a stop (still moving north)
C … Turn around; gradually speed back up again heading south
D … Continue heading south; gradually slow to a stop near the starting point

51. Tangent Lines t On a position vs. time graph: SLOPE VELOCITY Positivex
Tangent Lines t
On a position vs. time graph:
SLOPE
VELOCITY
Positive
Negative
Zero
SLOPE
SPEED
Steep
Fast
Gentle
Slow
Flat
Zero

52. Increasing & Decreasingt x
Increasing
Decreasing
On a position vs. time graph:
Increasing means moving forward (positive direction).
Decreasing means moving backwards (negative direction).

53. Acceleration and Velocity
As velocity increases, so does acceleration
As velocity decreases, so does acceleration
When direction changes, so does acceleration
When there is a constant velocity, there is no acceleration

54. Review

55. The Kinematic Equations describe the mathematical relationships that exist between an object’s motion.
Displacement
Velocity
Acceleration
Time

56. vf + vi 2 x = t Note that: Displacement with Constant Acceleration:
This equation does not require acceleration.
vf + vi 2
x = t

57. Example
A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find the distance that the car travels during braking. (Hint: list your givens)

58. Example
A car accelerates uniformly from rest to a speed of 6.6m/s in 6.5s. Find the distance the car travels during this time.

59. vf = vi + a∆t Note that: Velocity with Constant Acceleration:
This equation does not require displacement.
∆t = (tf – ti)
vf = vi + a∆t

60. Δx = vit + ½at2 Note that: “t” is actually “∆t”.
Displacement with Constant Acceleration:
Note that:
“t” is actually “∆t”.
However, ti is usually 0.
The equation can be rewritten to find xi.
Δx = vit + ½at2

61. Example
A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8m/s/s for 15s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

62. vf2 = vi2 + 2a∆x Note that: Velocity with Constant Acceleration:
This equation does not require time.
∆x = (xf – xi)
vf2 = vi2 + 2a∆x

63. Example
A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500m/s/s. What is the velocity of the stroller after it has traveled 4.75m?

64. Equations for Constantly Accelerated Straight-Line Motion

65. Practice a=11.2 m/s/s and d = 79.8
A race car accelerates uniformly from 18.5m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance it traveled.
a=11.2 m/s/s and d = 79.8

66. Falling Objects
free fall – downward acceleration while under the effect of gravity only.
Gravity is a force that causes objects to accelerate downward.

67. All objects free fall with an acceleration of g= -9.81 m/s2.
gravity causes objects to speed up as they fall downward and slow as they travel upward
we will usually neglect air resistance until FRICTION is covered in more detail.
For objects that are falling, launched, or thrown, downward acceleration is the value of gravity.

68. Graphing Free Fall
Curved line on a position vs time graph signifies an accelerated motion

69. Free Fall Practice Problems
Upton Chuck is riding the Giant Drop at Great America. If Upton free falls for 2.6 seconds, what will be his final velocity and how far will he fall?
D – 33m
Vf = 25.5 m/s

70. Free Fall Practice Problems
A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the moon is 1.67 m/s/s. Determine the time for the feather to fall to the surface of the moon.
T=1.29s

71. Bellringer 9/10
A tennis ball is thrown vertically upward with an initial velocity of +8.0m/s. What will the ball’s speed be when it returns to its starting point? How long will the ball take to reach its starting point?

72. Practice Problems
A race car traveling at 44 m/s accelerates to a velocity of 22 m/s over a period of 11 seconds.
What was the car’s displacement during this time?
360 m.

73. Practice Problems
A bike rider accelerates to a velocity of 7.5 m/s during 4.5 seconds.
If the bike had a displacement of 19 m, what was its initial velocity?
.9 m/s

74. Practice Problems
A stone is dropped into a deep well and is heard to hit the water 3.41s after being dropped. Determine the depth of the well.
-57.0

75. Practice Problems
Sketch a position time graph for the following scenarios:
An object moving with a constant positive velocity
An object at rest

76. Practice Problems
Sketch a velocity time graph for the following motions
A city bus that is moving with a constant velocity
A tiger that is speeding up at a uniform rate of acceleration while moving in the negative direction

77. Practice Problems
A train accelerates from a velocity of 21 m/s with an acceleration of 3 m/s2 over a distance of 535 meters.
What is the final velocity of the train?
60 m/s

78. Practice Problems
A worker drops a wrench from the top of a tower 80.0m tall. What is the velocity when the wrench strikes the ground?
-39.6 m/s

79. Practice Problems
A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0s.
What is the car’s average acceleration?
What is the car’s average velocity?
How far did the car move while accelerating?