1. In this section we will…
…develop our understanding of using numbers and equations to describe motion.

2. What equations to describe motion have you come across before?
Think and Talk
What equations to describe motion have you come across before?

3. What do we understand about 'acceleration'?
Think and Talk
What do we understand about 'acceleration'?

4. acceleration (m s–2) is

5. Rearrange this to give v =…
rate of change of velocity per unit time
Rearrange this to give v =…

6. v = u + at

7. The Equations of Motion
There are three equations which together are known as the equations of motion.

8. The Equations of Motion?
When can I use…
The Equations of Motion?
when acceleration is constant (uniform)
and
motion is in a straight line

9. The Equations of Motion?
Do I have to learn…
The Equations of Motion?
You need to be able to:
 select the correct formula
 identify symbols and units
 carry out calculations to solve the problems of real-life motion.

10. The Equations of Motion?
Do I have to learn…
The Equations of Motion?
You need to be able to:
 carry out experiments to verify the equations of motion

11. To do this fully, you might find it an interesting challenge to…
 understand where the equations come from.

12. The Equations of Motion
v = u + at
Label the formula using correct symbols and units

13. Equation 2
Describe the motion of this object

14. Equation 2
How can we determine the displacement of the object?

15. Equation 2 Area under the graph = 1500 + 4500 = 6000 m Area 2 = ½bh
= ½ × (35 –- 5) × 300
= 4500 m
Area 1 = 5 × 300 = 1500 m

16. Equation 2 v Area 2 = ½bh = ½ × (v – u) × t Since v = u + at
and v – u = at
Area 1 = ut t

17. Equation 2 Area under the graph = displacement s v Area 2 = ½bh
= ½ × (v – u) × t
= ½ × at × t
= ½ × at2
Area 1 = ut t

18. Equation 2 s = ut + ½at2 v Area 2 = ½bh = ½ × (v – u) × t = ½ × at × t
Area 1 = ut t

19. Equation 3
Start with Equation 1
v = u + at

20. Equation 3
and square it
v2 = (u + at)2

21. v2 = u2 + 2uat + a2t2 v2 = u2 + 2a(ut+ ½at2)
Equation 3
v2 = u2 + 2uat + a2t2 v2 = u2 + 2a(ut+ ½at2)
Equation 2!

22. Equation 3
v2 = u2 + 2as

23. Using the Equations of Motion
What do we need to think about when using the equations of motion?
Symbols, units. Standard techniques for layout.
Vector quantities and direction.

24. velocity displacement
Using the
Equations of Motion
What do the following quantities have in common?
velocity displacement
acceleration
Vector quantities and therefore positive direction must be chosen and the reverse direction given a negative sign.

25. The sign convention When dealing with vector quantities we
must have both magnitude and
direction.
When dealing with one-dimensional
kinematics (ie motion in straight lines) we
use + and – to indicate travel in opposite
directions. L

26. The sign convention Normally we use the following convention:
negative –
positive +
negative –
positive +

27. Take care – in some questions the sign convention is reversed
Normally we use the following convention
negative –
positive +
negative –
positive +
Take care – in some questions the sign convention is reversed

28. v = u + at Equation 1 and the Sign Convention
What does a positive value of acceleration mean?

29. Using the normal sign convention

30. –ve +ve Christine Arron is a 100-m sprint athlete.
Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s.
Her maximum velocity is
10.49 m s–1.
Calculate her acceleration over the first 50 m of the race, showing full working.
Christine Arron is a track and field sprint athlete, who competes for France.
Statistics on elite athletic performance quoted from
v = u + at
10.49 = 0 + a(4.16)
a = 2.52 m s–2

31. –ve +ve Her acceleration is 2.52 m s–2.
In this case, acceleration is a rate of change of velocity with time, with which we are familiar.
A positive value means, in this case, increasing velocity with time.
What else might it mean?
Christine Arron is a track and field sprint athlete, who competes for France.
Statistics on elite athletic performance quoted from
v = u + at
10.49 = 0 + a(4.16)
a = 2.52 m s–2

32. –ve +ve As she passes the finish line, Christine begins to slow down.
She comes to rest in 8.20 s from a velocity of 9.73 m s–1.
Calculate her acceleration, showing full working.
Statistics on elite athletic performance quoted from
v = u + at
0 = a(8.20)
a = –1.19 m s–2

33. –ve +ve Her acceleration is a = –1.19 m s–2.
Notice that the acceleration has a negative value.
Explain this.
Acceleration with a negative value: in this case losing speed in the positive direction.

34. –ve +ve Now consider Christine running in the opposite direction.
Notice that the sign convention remains the same.
Christine Arron is a track and field sprint athlete, who competes for France.
Statistics on elite athletic performance quoted from
v = u + at
–10.49 = 0 + a(4.16)
a = –2.52 m s–2

35. –ve
+ve
Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s.
Her maximum velocity is
–10.49 m s–1 (why is it negative?).
Calculate her acceleration over the first 50 m of the race, showing full working.
Christine Arron is a track and field sprint athlete, who competes for France.
Statistics on elite athletic performance quoted from
v = u + at
–10.49 = 0 + a(4.16)
a = –2.52 m s–2

36. –ve +ve Her acceleration is –2.52 m s–2. What does the negative mean?
Acceleration with a negative value: in this case gaining speed in the negative direction.

37. –ve +ve As she passes the finish line, Christine begins to slow down.
She comes to rest in 8.20 s from a velocity of –9.73 m s–1.
Calculate her acceleration, showing full working.
Statistics on elite athletic performance quoted from
v = u + at
0 = – a(8.20)
a = 1.19 m s–2

38. –ve +ve Her acceleration is a = 1.19 m s–2.
Notice that the acceleration has a negative value.
Explain this.
Acceleration with a positive value: in this case losing speed in the negative direction.

39. Equation 1 and the sign convention
A positive value means gaining speed while moving in the positive direction.
–ve
+ve
Initial velocity
Final velocity
Acceleration
0 m s–1
m s–1

40. OR
A positive value means the object is losing speed while moving in the negative direction.
–ve
+ve
Initial velocity
Final velocity
Acceleration
–9.73 m s–1
0 m s–1

41. In summary:
A negative value means the object is gaining speed while moving in the negative direction.
–ve
+ve
Initial velocity
Final velocity
Acceleration
–10.49 m s–1
0 m s–1

42. OR
A negative value means the object is losing speed while moving in the positive direction.
–ve
+ve
Initial velocity
Final velocity
Acceleration
0 m s–1
m s–1

43. –ve
+ve
Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist.
Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s–1 in 1.75 s. Calculate his displacement in this time.

44. Using the Equations of Motion
Step 1: Write down the sign convention.
Step 2: Write down what you know (think suvat).
s displacement
u initial velocity
v final velocity
a acceleration
t time
Step 3: Any other information,
eg acceleration due to force of gravity?

45. Using the Equations of Motion
Step 4: Select formula – use data sheet.
Step 5: Substitute values then rearrange formula.
Step 6: Write the answer clearly, including magnitude and direction, and units.

46. –ve
+ve
Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist.
Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s-1 in 1.75 s. Calculate his displacement in this time.

47. –ve
+ve
s = ? m
u = 0 m s–1
v = 8.70 m s–1
a = ?
t = 1.75 s

48. -ve
+ve
s = ? m
u = 0 m s–1
v = 8.70 m s–1
a = ?
t = 1.75 s

49. -ve
+ve
s = ? m
u = 0 m s–1
v = 8.70 m s–1
a = ?
t = 1.75 s

50. -ve
+ve
s = ? m
u = 0 m s–1
v = 8.70 m s–1
a = ?
t = 1.75 s

51. In the previous section we developed…
…our understanding of using graphs to describe motion …our skills in interpreting graphs of motion …our skills in describing motion using physics terms correctly.
Written to follow on from ODU 1c).

52. In this section we planned to…
…develop our understanding of using numbers and equations to describe motion.

53. Next we will bring all of this together and use…
…our understanding of using graphs to describe motion
…our skills in interpreting graphs of motion
…our skills in describing motion using physics terms correctly
… our understanding of using numbers and equations to describe motion
for vertical motion

54. Everyday acceleration
What sort of accelerations do you experience in everyday life? How can this be investigated?
Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc.
Images from Microsoft Clip Art.

55. Everyday acceleration
Do you experience accelerations only in the horizontal?
Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc.
Images from Microsoft Clip Art.

56. Everyday acceleration
An accelerometer (a device which measures acceleration in three dimensions) can be used to investigate accelerations which you experience in everyday life.
Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc.
Images from Microsoft Clip Art.

57. A stationary tennis ball
Holding the ball stationary, describe its motion.
Images from Microsoft Clip Art.
Describe its motion.

58. A tennis ball travelling vertically upwards
Describe its motion.
Film the ball as it is thrown upwards
and use tracker.jar to analyse its motion.
Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement.
Link to tracker.jar tutorial lesson.
Images from Microsoft Clip Art.

59. A tennis ball dropped from a height
Describe its motion.
Film the ball as it falls and use tracker.jar to analyse its motion.
Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement.
Link to tracker.jar tutorial lesson.
Images from Microsoft Clip Art.

60. Two tennis balls dropped from a height
Predict the motion.
Observe.
Explain!
The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects.
Images from Microsoft Clip Art.

61. Two tennis balls dropped from a height
Was your initial prediction that the two identical tennis balls dropped from the same height would hit the ground at exactly the same time?
The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects.
Images from Microsoft Clip Art.

62. Two tennis balls dropped from a height
What did you think when you discovered that one ball had a significantly greater mass than the other?
What do you think should have happened?
What did you observe?
The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects.
Images from Microsoft Clip Art.

63. The mass does not matter!
Both balls will hit the ground at the same time when dropped from the same height.
If you do not believe this,
tracker.jar will allow you to analyse the motion.
The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects.
Images from Microsoft Clip Art.

64. © Erich Schrempp / Science Photo Library

65. © Nicola Jones

66. The elephant and the feather in free-fall
Suppose we can
switch off air
resistance.
Which will hit
the ground first?
Images from Clipart

67. The elephant and the feather with air resistance
The force of gravity near
the Earth’s surface gives
all objects the same
acceleration.
So why doesn’t the
feather reach the ground
at the same time as the
elephant?
Images from Clipart

68. Dropping an elephant…
We commonly use
a negative to
indicate
downward motion.
Images from Clipart
but be warned – you may come across questions in which the sign convention is reversed.

69. Dropping an elephant in the absence of air resistance.
Images from Clipart

70. Dropping an elephant in the absence of air resistance
Calculate speed, velocity, distance and displacement at 1-s intervals.
Time
(s)
Speed
(m s–1)
Velocity
Distance fallen (m)
Displacement (m) 1 2 3 4 5
Print off to save time!!!

71. Sketch graphs to show how the speed, distance, velocity and displacement vary with time during the free-fall.
Speed
Distance
Time
Time
Velocity
Displacement
Time
Time

72. What is missing from the graph sketches?
Sketch graphs to show how the speed, distance, velocity and displacement vary with time during free-fall.
What is missing from the graph sketches?
Speed
Distance
Time
Time
Velocity
Displacement
Time
Time

73. Speed (m s–1)
Distance ( m)
Time (s)
Time (s)
Velocity (m s–1)
Displacement (m)
Time (s)
Time (s)

74. A tennis ball dropped from a height and allowed to bounce
Consider the ball being dropped, allowed to bounce and return to its original height.
Sketch your predictions for speed–time, velocity–time and acceleration–time graphs.

75. Compare to the results from this simulation.

76. Describe the motion.

77. When dropped, the ball gains speed in the negative direction hence the –ve sign for acceleration.
The ball then loses speed in the positive direction, coming to rest at the original height.
Does this happen in real life? Explain!

78. a ( m s–2)
Time (s)

79. Consider a tennis ball thrown upwards and allowed to fall back to its starting position.

80. A tennis ball thrown upwards then allowed to fall back to its starting position
Sketch the velocity, speed and
acceleration graphs to describe its motion until it returns to its starting position.

81. Virtual Higher Experiments → Higher Physics → Mechanics and Properties of Matter → Activity 5b.

82. What is the force acting on a tennis ball thrown upwards?

83. Estimate the initial acceleration of a jumping popper.

84. Calculate the initial acceleration of a jumping popper.
What assumptions are you making?
How could your calculation be improved?

85. Observe what happens when the groaning tube is dropped.
Explain!

86. In the previous section we developed…
…our understanding of using graphs to describe motion …our skills in interpreting graphs of motion …our skills in describing motion using physics terms correctly.
Written to follow on from ODU 1c).

87. Review your progress! In this section we…
…developed our understanding of using numbers and equations to describe motion.
Review your progress!