1. Picturing Motion on a Cartesian Graph

2. You can also describe motion with a
Velocity vs Time Graph

3. Car “A”
Car “B”
Describe what is occurring
Car “A” is moving at a constant velocity of +11 m/s
Car “B” is moving at a constant velocity of -33 m/s

4. Describe the motion positive velocity (slowing)
Positive velocity (constant)
Describe the motion

5. Describe the motion negative Positive velocity( slowing) stopped
Negative velocity,(speeding up)
Describe the motion
negative
Is the acceleration positive or negative?

6. Stopped (0 m/s)
negative velocity (slowing)
Describe the motion

7. 10 m/s - 0 m/s = .25 m/s2 40 s - 0 s positive velocity (speeding up)
Describe what is occurring
10 m/s - 0 m/s
40 s - 0 s
= .25 m/s2
What is the slope of this line?

8. graph is the acceleration. 10 m/s - 0 m/s 40 s - 0 s = .25 m/s2
The slope of a V vs T
graph is the acceleration.
10 m/s - 0 m/s
40 s - 0 s
= .25 m/s2
Note that the acceleration is POSITIVE

9. Is the acceleration constant?
Yes, because the slope is constant!!!

10. What does this graph tell you about the two objects?

11. Applets for graphs 2.5

12. Can do WS 2.3 (Motion Graph Interpretation WS)
The one with lots of pictures of graphs

13. Positive velocity, speeding up at an exponentially
Describe the motion

14. What is the average acceleration over 90 seconds?
v2 - v1
t2 - t1 =
48 m/s -
3 m/s
90 s -
0 s
=.50 m/s2

15. How would you find instantaneous acceleration?
Tangent Lines

16. Finding Displacement from a velocity time graph.Dx Dt = Dx
= v Dt

17. 50 m/s
The area under a V vs T line is the displacement
30 s
Find the displacement from time = 40 seconds to time = 70 sec Dx
= v Dt
= 50 m/s 30 s

18. Dx = v Dt Find the displacement from time = 0 s to 30 sec v2 = 30 m/s
Using the equation below
And by finding the area under the line Dx
= v Dt

19. +900 m
-1200 m
What is the displacement from 0 – 70 s?
-300 m

20. Could we find the displacement here?
Just break it up into smaller problems

21. OR

22. Find the displacement between the two pointsv1 v2
Describe the scenario
Find the displacement between the two points

23. Find the displacement between the two pointsv2
Area 2 v1
Area 1 t1 t2
Find the displacement between the two points
Displacement = Area 1 + Area 2

24. v2
Area 2 v1
Area 1 t1 t2
Area 1 =
V1(t2-t1) =
v1Dt

25. v1Dt (t2-t1) (v2-v1) Area 2 Area 2 = 1/2 base * height = 1/2
= 1/2 Dt (v2-v1)

26. Displacement = v1Dt + 1/2 Dt (v2-v1)
height v1
base
v1Dt t1 t2
Displacement = v1Dt + 1/2 Dt (v2-v1)

27. Displacement = v1Dt + 1/2 Dt (v2-v1)
Dx = Dt(v1 + 1/2 (v2-v1)) Dx
= (v1 + 1/2 (v2-v1)) Dt Dx
= v1 + 1/2 v2- 1/2 v1 Dt Dx
= v1 - 1/2 v1 + 1/2 v2 Dt

28. Dx
= v1 - 1/2 v1 + 1/2 v2 Dt Dx
= 1/2 v1 + 1/2 v2 Dt Dx
= 1/2 (v1 + v2) Dt Dx
(v1 + v2)
v = = Dt 2

29. This works for any situation where a triangle is formed
our shapes fit the line v v t t v v t
What is different about acceleration? t
What is different about acceleration?

30. Shortcut ONLY WORKS if acceleration is constant.
Meaning velocity is a straight line.
(v1 + v2)
v = 2 Dx
v = Dt
Is always TRUE!!!
It is true by definition.

31. What is the average velocity from 0 to 60 seconds?

32. How do would you find
Initial position
displacement from 20 to 60 seconds
instantaneous speed at 30 seconds
average speed from 0 to 40 seconds
acceleration

33. How do would you find
Initial position
displacement from 20 to 60 seconds
instantaneous speed at 50 seconds
average speed from 0 to 70 seconds
average acceleration from 10 to 90 seconds

34. How do would you find
Initial position
displacement from 20 to 60 seconds
instantaneous speed at 50 seconds
average speed from 0 to 70 seconds
average acceleration from 10 to 90 seconds

35. Initial position
displacement from 0 to 80 seconds
average speed from 0 to 80 seconds
instantaneous speed at 0 seconds
instantaneous speed at 80 seconds
Acceleration from 0 to 80

36. Initial position
displacement from 0 to 90 seconds
average speed from 0 to 90 seconds
instantaneous speed at 0 seconds
instantaneous speed at 90 seconds
Acceleration from 0 to 90

37. Motion Graph WS
Calculations with motion graphs WS

38. Equations SO FAR Dx
v = Dt
(v2 + v1)
v = 2 Dv
a = Dt

39. A car accelerates uniformly from rest to 48 m/s in 8. 9 seconds
A car accelerates uniformly from rest to 48 m/s in 8.9 seconds. How far did it travel in this time? Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

40. A roller coaster starts from rest and accelerates at 6
A roller coaster starts from rest and accelerates at 6.8 m/s2 for 18 seconds. How far does it travel? Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

41. A car initially moving at 13 m/s accelerates and travels 120 m in 4
A car initially moving at 13 m/s accelerates and travels 120 m in 4.5 seconds. What was its final speed? (assuming a constant acceleration) Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

42. A car starts from rest and reaches 60 m/s in 45 meters
A car starts from rest and reaches 60 m/s in 45 meters. What was its acceleration? Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

43. What was the initial speed of a car which finishes a 600 m race at 65 m/s, if it did so in 38 seconds? Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

44. A car starts from rest and accelerates at 3. 5 m/s2
A car starts from rest and accelerates at 3.5 m/s2. How far has it gone when it reaches 25 m/s? Dx
v = Dt
(v2 + v1) 2
v = Dv
a = Dt

45. Motion Problems WS

46. Derivation Dx
v = Dt
(v2 + v1)
Dx = v1t + at2 2
v = 2 Dv
a = Dt

47. Dx = v1t + at2 2 Time (s) Change in time Displacement(m)
Initial velocity(m/s)
Acceleration (m/s2)

48. What acceleration would cause a car starting from a dead stop at 220 m to reach 110 m in 30.0 seconds?
a = ??
Dx = v1t + at2
v0 = 0 m/s 2
d0 = 220 m
df = 110 m
220 m
110 m
30 s
t = 30 s

49. at2
Dx = v1t + 2
Applets for graphs
Exploration 2.4

50. Initial velocity(m/s)
The last big one, which variable is left out?
v2 = v a Dx 2 2
displacement (m)
Acceleration (m/s2)
Final Velocity(m/s)
Initial velocity(m/s)

51. v2 = v1 + 2a(x2 - x1) a = 2.5 m/s2 df = 800 m do = 0 m vf = 64 m/s
A car accelerates at a constant rate of 2.5 m/s2 and finishes an 810 m straight track and reaches a final velocity of 64 m/s. What was its initial velocity?
v2 = v1 + 2a(x2 - x1) 2 2
a = 2.5 m/s2
df = 800 m
do = 0 m
810 m
0 m
vo = ?
vf = 64 m/s
vf = 64 m/s
vo = ?????

52. Dx Dv v = a = Dt Dt (v2 + v1) v = 2 at2 Dx = v1t + 2 v2 = v1 + 2 a Dx
YOUR
ARSENAL Dx Dv
v =
a = Dt Dt
of equations
(v2 + v1)
v =
of equations 2
a is average acceleration (or constant)
at2
Dx = v1t + 2
t is change in time
v2 = v a Dx 2 2
UNITS NEED to match

53. a = 9.8 m/s2 x2 = 15.3 cm =.153 m x1 = 0 m t = ???? v1 = 0 m/s
A person drops a ruler from rest which accelerates at 9.8 m/s2 . The ruler is caught after 15.3 cm of travel. How long did it take the person to catch it?
v2 = v1 + 2aDx 2
Dx = v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =
a = 9.8 m/s2
x2 = 15.3 cm
=.153 m
x1 = 0 m
t = ????
v1 = 0 m/s

54. A car starts at 74 m with an initial velocity of -35 m/s and accelerates at 6.5 m/s2 for 23 seconds. What is its final position?
v2 = v1 + 2aDx 2
Dx = v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =

55. What is the final speed of a meteorite which enters our atmosphere at 150 m/s and accelerates at -4.6 m/s2 through the atmosphere which is 1500 m thick.
v2 = v1 + 2aDx 2
Dx = v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =

56. How long will it take to travel 524 m at an average velocity of 12
How long will it take to travel 524 m at an average velocity of 12.4 m/s
v2 = v1 + 2aDx 2
Dx = v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =

57. v0 = 0 m/s vf = 15.6 m/s t = 61 s df = ??? d0 = 0 m
A car starts from rest and accelerates to 15.6 m/s in 61 seconds how far did it travel in this time?
v2 = v1 + 2aDx 2
Dx = v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =
v0 = 0 m/s
vf = 15.6 m/s
t = 61 s
df = ???
d0 = 0 m
Or use could have used

58. Problems page 43 19 – 26, 29 Read section 2-7 2.2 m/s2, 110 m
4.41 m/s2, 2.61 s
62.5 m
33 m/s
I would try reading and trying the question on page
(these are like multiple choice questions)
160 m, 25 s, 12 m, 10 m
-390 m/s2
s , 240 km/h
Read section 2-7

59. Gravity accelerates ALL objects in free fall at
g = 9.80 m/s2
If gravity is the only force acting on an object (no wind resistance etc…)

60. 1 kg and 5 kg hunks of lead are dropped from a building
Which hits the ground first?

61. If they both accelerate at 9.8 m/s2 they hit at the same time!!!

62. But does this really happen???
Do heavy object fall at the same rate as light ones???

63. Objects accelerate the same NO matter the mass?
If I drop a rock and piece of paper will they fall at the same rate?

64. Why don’t all things fall at the same rate on earth
Air resistance

65. If we could drop stuff in an environment without air
we could test this theory.
Lunar drop

66. g = 9.80 m/s2 Gravity accelerates objects in free fall at
Should g be positive or negative?

67. Freefall speed of objects dropped from rest
Instantaneous Speed
(m/s)
Time Falling
(s) 1
9.8 2
19.6 3
29.4
9.8t t

68. What would the velocity of an object in
freefall look like graphed
What is the initial acceleration when dropped from rest?

69. Instantaneous velocity
0 m/s 1s
9.8 m/s
How fast is the rock moving after 1 second? 2s
19.6 m/s
Are these its
average velocity or
Instantaneous velocity 3s
29.4 m/s
30.2 m/s 4s

70. How far did the rock fall during the 1st second of freefall
0 m/s
9.8 m/s
19.6 m/s
29.4 m/s
30.2 m/s
How far did the rock fall during the 1st second of freefall
40 m/s

71. Why does the distance traveled
0 m/s
9.8 m/s
19.6 m/s
29.4 m/s
30.2 m/s
5 m
15 m
= ?
Why does the distance traveled
each second
increase each second?
= ?
25 m
35 m
= ?

72. Distance fallen changes with t2

73. How much does the velocity of an object increase each second of fall?
How much does the Displacement of an object increase each second of fall?

74. v = gt
gt2 2
Dy =

75. There are 3 main types of problems
When working “gravity” problems
acceleration is a given (9.80 m/s2)
There are 3 main types of problems
Throwing
upward
Starting
from rest
Throwing down

76. v1 = 0 m/s v2 = ??? t = 2.5 sec a = 9.80 m/s2 Dv = at
If I drop a rock (from rest) off a building, what is its velocity after 2.5 seconds? How far has it moved?
Draw a picture
v1 = 0 m/s
Define coordinates
v2 = ???
Establish what you know
Pick the equation(s)
t = 2.5 sec
a = 9.80 m/s2 +
Dv = at
Dv = 9.80 m/s2 2.5s
v2 = 24 m/s
How far has it traveled?

77. v1 = 0 m/s v2 = 25 m/s t = 2.5 sec a = 9.80 m/s2 v Dt Dy =
If I drop a rock (from rest) off a building, what is its velocity after 2.5 seconds? How far has it moved then?
v1 = 0 m/s
How far has it traveled?
1 way to solve is:
v2 = 25 m/s
t = 2.5 sec
Find average velocity
a = 9.80 m/s2 +
v =
0 m/s + 24 m/s 2
= 12 m/s
You now know average velocity and time
Dy =
v Dt
= 12 m/s 2.5 s = 30 m

78. at2 Dy = v1t + 2 How far has it traveled? v1 = 0 m/s t = 2.5 sec
Another strategy
t = 2.5 sec
at2
a = 9.80 m/s2
Dy = v1t + 2

79. A rock is dropped from rest, 6. 0 meters above the ground
A rock is dropped from rest, 6.0 meters above the ground. How fast is it going when it hits the ground?
v1 = 0 m/s
Draw a picture
y2 = 6 m
Define coordinates
y1 = 0 m
Establish what you know
Pick the equation(s)
a = 9.80 m/s2 Dy Dv
of equations
vf = ????
6 m
v =
a = + Dt Dt
of equations
(v1 + v2) 2
v =
Dy= v1t + at2 2
v2 = v1 + 2a(y2 - y1) 2 2

80. vf = vo + 2a(y2 – y1) v0 = 0 m/s y2 = 6 m y1 = 0 m a = 9.80 m/s2
A rock is dropped from rest, 6.0 meters above the ground. How fast is it going when it hits the ground?
v0 = 0 m/s
Draw a picture
y2 = 6 m
Define coordinates
y1 = 0 m
Establish what you know
Pick the equation(s)
a = 9.80 m/s2
vf = ????
6 m +
vf = vo + 2a(y2 – y1) 2
vf2 = (0 m/s)2 + 2 · 9.80 m/s2 · 6.0 m
vf = 11 m/s (down)

81. v1 = 0 m/s t = 4.6 sec a = 9.80 m/s2 y2 = ??? y1 = 0 m
You drop a rock (from rest) and hear it hit 4.6 s later. How high are you up?
v1 = 0 m/s
t = 4.6 sec
a = 9.80 m/s2 +
y2 = ???
y1 = 0 m
v2 = v1 + 2a(y2 - y1) 2
y2 = y1 + v1t + at2
of equations
(v1 + v2)
v = Dy Dt Dv
a =

82. v0 = 0 m/s t = 4.6 sec a = 9.80 m/s2 df = ??? d0 = 0 m
You drop a rock (from rest) and hear it hit 4.6 s later. How high are you up?
v0 = 0 m/s
t = 4.6 sec
a = 9.80 m/s2 +
df = ???
d0 = 0 m
y2 = y1 + vot + at2 2
y2 = 0 m + 0 m/s 4.6s m/s2 (4.6 s)2
y2 = 210 m

83. at2 Dy= v1t + 2 If you want to know how high you are.
Just drop an rock.
Count 1 Mississipi, etc…
your height (in m) is just
9.8(time)2
if you don’t have a calculator approximate
at2
Dy= v1t + 2 y= 2
You drop and rock and count 5 Mississippi’s
How high are you?
about 250 meters ( about 750 feet)

84. Will the object speed up or slow down?
Throwing downward
Will the object speed up or slow down? v1 g
What are the signs on v1 and g?

85. If you throw a rock down at 6.4 m/s from a height of 34 meters.
How fast is it going when it hits the ground?
v2 = v1 + 2a(y2 - y1) 2
x2 = x1 + v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =

86. Throwing upward
Will the object speed up or slow down? vo g
What are the signs on v1 and g?

87. If I toss a rock upwards at 30 m/s, what will happen to its speed?
For simplicity, I am going to round gravity to 10 m/s2 3s
0 m/s 2s 4s
10 m/s 1s
20 m/s 5s 6s
30 m/s
40 m/s 7s

88. Is the velocity the same as when it left?
0 m/s
The speed is the same
when it hits the ground
as….. 2s 4s
10 m/s 1s
20 m/s 5s
Is the velocity the same
as when it left?
30 m/s 6s
What does this tell you about shooting a gun into the air?

89. What is the acceleration here?
0 m/s 2s 4s
10 m/s 1s
20 m/s 5s
30 m/s 6s

90. For an object thrown upward
Where is the highest point?
What is its acceleration then?

91. Key things to remember about this.
The speed at the highest point is…. 3s
0 m/s 2s 4s
10 m/s
The speed when it reaches the ground again is… 1s
20 m/s 5s
The time it takes to fall back down is equal to….
30 m/s 6s
The time it takes to reach its highest point is _____
Of the total round trip time

92. If you throw a rock upward at 64 m/s.
How long does it take to reach its highest point?
How long after it hits the highest point does it take to reach your hand again?
v2 = v1 + 2a(y2 - y1) 2
y2 = y1 + v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =
Total round trip time?
How high did it go?
What was its velocity when it hit your hand again?
How fast is it going if it lands 35 meters below your hand?

93. Dy= v1t + at2 2 2.0 m = 29 m/s t + -4.9 m/s2 t2
If you throw a rock upward at 29 m/s.
At what time is it 2.0 meters above your hand?
Dy= v1t + at2 2
v2 = v1 + 2a(y2 - y1) 2
y2 = y1 + v1t + at2
of equations
(v1 + v2)
v = Dx Dt Dv
a =
2.0 m = 29 m/s t m/s2 t2
Rearrange
0 = -4.9 t t

94. 0 = a x2 + b x + c 0 = -4.9 m/s2 t2 + 29 m/s t -2 m x =
t = .069 sec, and 5.8 sec
Which one is right?

95. .069 sec
5.8 sec
2 m
Time not to scale but you get the point?

96. Key things to remember about throwing upward problems
the time to go UP is the SAME as the time going down
IF…..
so the round trip is twice the time it takes to reach v=0
the speed at which it hits the ground is the same as the speed that it left (is the velocity?)

97. Honors Physics: Page 44 Problems 34 – 38, 41,44,46 34 60.0m
Acceleration due to gravity problems:
Honors Physics:
Page Problems 34 – 38, 41,44,46 m
s, 86.3 m/s down
36 32m, 5.1 s s
38 13m s
44 +/ m/s, s, s m

98. Few review questions

99. A slingshot launches a marble into the air, and it hits the ground
5.2 s later.
Find: Time ascending
Time descending
highest position
initial velocity

100. A bullet is fired straight up into the air at 440 m/s
A bullet is fired straight up into the air at 440 m/s. When does it hit the ground?

101. An astronaut drops a hammer from a height of 1
An astronaut drops a hammer from a height of 1.6 m, and it hits the aliens planet’s surface 2.5 seconds later. What is the acceleration due to gravity on this planet.

102. A rock is dropped from a building, After 3 seconds, another identical rock is dropped from the same spot. What happens to the distance between the rocks while both are in free fall? ?

108. Few review questions

109. A slingshot launches a marble into the air, and it hits the ground
5.2 s later.
Find: Time ascending
Time descending
highest position
initial velocity