1. Section 3.7 Calculus AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us
Particle Motion
Section 3.7
Calculus AP/Dual, Revised ©2013
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3.7 – Particle Motion

2. What is Position, Velocity, and Acceleration?
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3.7 – Particle Motion

3. Motions
Position known as s(t) or x(t); also known as speed is the rate of motion
Label could be known as meters
“Initially” means when t = 0
“At the origin” means x(t) = 0
Velocity known as v(t) = s’(t); abs value rate of motion or known as SPEED and DIRECTION
Label could be known as meters/second or speed/time
“At rest” means v(t) = 0
If the velocity of the particle is positive, then the particle is moving to the right
If the velocity of the particle is negative, then the particle is moving to the left
If the order of the particle changes direction, the velocity must change signs
Acceleration known as a(t) = v’(t) = s’’(t)
Label could be known as meters/second2 or velocity/time
If the acceleration of the particle is positive, then the particle is increasing
If the acceleration of the particle is negative, then the particle is decreasing
If a particle slows down, signs from v’(t) and s’’(t) are different (SIGNS DIFFERENT)
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3.7 – Particle Motion

4. Formulas
Displacement is defined as the shortest distance from the initial to the final position of a point
Time interval of (a, b)
Equation: s(b) – s(a)
Total distance measures distance from forward and backwards
Average Velocity = 𝒔 𝒃 −𝒔 𝒂 𝒃−𝒂 or divide the change in position by the change in time
Instantaneous Speed = 𝒗 𝒕
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3.7 – Particle Motion

5. Graph
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3.7 – Particle Motion

6. Website
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3.7 – Particle Motion

7. Geometric Sketchpad
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3.7 – Particle Motion

8. Example 1a
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟎 𝒕 𝟐 +𝟓 where t is measured in seconds and s is in meters. Determine the a) position, b) instantaneous velocity, c) acceleration and d) speed of the particle all at t = 1.
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3.7 – Particle Motion

9. Example 1b
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟎 𝒕 𝟐 +𝟓 where t is measured in seconds and s is in meters. Determine the a) position, b) instantaneous velocity, c) acceleration and d) speed of the particle at t = 1.
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3.7 – Particle Motion

10. Example 1c
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟎 𝒕 𝟐 +𝟓 where t is measured in seconds and s is in meters. Determine the a) position, b) instantaneous velocity, c) acceleration and d) speed of the particle at t = 1.
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3.7 – Particle Motion

11. Example 1d
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟎 𝒕 𝟐 +𝟓 where t is measured in seconds and s is in meters. Determine the a) position, b) instantaneous velocity, c) acceleration and d) speed of the particle at t = 1.
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3.7 – Particle Motion

12. Example 2a Velocity at time t
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
Velocity at time t
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3.7 – Particle Motion

13. Example 2b Acceleration
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
Acceleration
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3.7 – Particle Motion

14. Example 2c
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
At Rest
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3.7 – Particle Motion

15. Moving furthest from the Left (Relative MINIMUM)
Example 2d
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
Moving furthest from the Left (Relative MINIMUM)
X = 0
(0, 1/3)
X = 1/3
(1/3, 6)
X = 6
(6, ∞)
f(0)
(0, -7)
f ’(1/4)
(-)(-)
POSITIVE
RIGHT
f(1/3)
(1/3, -136/27)
Rel MAX
f ’(2)
(+)(-)
NEGATIVE
LEFT
f(6)
(6, 187)
Rel MIN
f ’(7)
(+)(+)
X = 0
(0, 1/3)
X = 1/3
(1/3, 6)
X = 6
(6, ∞)
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3.7 – Particle Motion

16. Example 2e Moving to the RIGHT
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
Moving to the RIGHT
X = 0
(0, 1/3)
X = 1/3
(1/3, 6)
X = 6
(6, ∞)
f(0)
(0, 7)
f ’(1/4)
(-)(-)
POSITIVE
RIGHT
f(1/3)
(1/3, -136/27)
Rel MAX
f ’(2)
(+)(-)
NEGATIVE
LEFT
f(6)
(6, 187)
Rel MIN
f ’(7)
(+)(+)
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3.7 – Particle Motion

17. SLOWING DOWN (get critical values and POI)
Example 2f
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
SLOWING DOWN (get critical values and POI)
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3.7 – Particle Motion

18. Example 2g
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
X = 0
(0, 19/6)
X = 19/6
(19/6, ∞)
f(0)
(0, 7)
f ” (x) = 12t – 38
f ” (1)
(-)
NEGATIVE
f ” (19/6)
f ”(4)
(+)
POSITIVE
X = 0
(0, 1/3)
X = 1/3
(1/3, 6)
X = 6
(6, ∞)
f(0)
(0, 7)
f ’(1/4)
(-)(-)
POSITIVE
RIGHT
f(1/3)
(1/3, -136/27)
Rel MAX
f ’(2)
(+)(-)
NEGATIVE
LEFT
f(6)
(6, 187)
Rel MIN
f ’(7)
(+)(+)
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3.7 – Particle Motion

19. Total distance at t = 3 secs?
Example 2h
The position function of a particle moving on a straight line is 𝒔 𝒕 =𝟐 𝒕 𝟑 −𝟏𝟗 𝒕 𝟐 +𝟏𝟐𝒕−𝟕 where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration, c) at rest, d) particle moving furthest from the left, e) particle moving to the right, f) slowing down, and g) total distance at t = 3 secs?
Total distance at t = 3 secs? t
s(t) –7
1/3
–5.037 3
–88
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3.7 – Particle Motion

20. Your Turn
The position function of a particle moving on a straight line is 𝒙 𝒕 =𝟑 𝒕 𝟒 −𝟏𝟔 𝒕 𝟑 +𝟐𝟒 𝒕 𝟐 from [–5, 5] where t is measured in seconds and x is in feet. Determine the a) velocity at time t, b) acceleration at time t, c) at rest, d) particle changing direction, and e) identify the velocity when acceleration is first zero.
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3.7 – Particle Motion

21. Speed, Velocity, and Tangent Lines
Speed is the absolute value of velocity. It is measured of how fast something is moving with the regard of direction
The effect of how an absolute value function has it on the graph is that it reflects all values that are below the x-axis, above on the x-axis
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3.7 – Particle Motion

22. Example 3
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
Time
Velocity 1 2 3 4 8 5 16
Speed 1 2 4 8 16
Speed
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2.2A - Rate of Change

23. Example 3
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
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2.2A - Rate of Change

24. Example 4
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
Time
Velocity 1 –1 2 –2 3 –4 4 –8 5
–16
Speed 1 2 4 8 16
Speed
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2.2A - Rate of Change

25. Example 4
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
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2.2A - Rate of Change

26. Example 5
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
Time
Velocity 1
–16 2 –8 3 –4 4 –2 5 –1
Speed 16 8 4 2 1
Speed
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2.2A - Rate of Change

27. Example 5
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
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2.2A - Rate of Change

28. Example 6
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
Time
Velocity 1 16 2 8 3 4 5
Speed 16 8 4 2 1
Speed
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2.2A - Rate of Change

29. Example 6
For each situation, the graph is differentiable when giving velocity as a function of time [1, 5] along the selected values of the velocity. In this graph, each horizontal mark represents 1 unit and each vertical mark represents 4 units. Plot the speed graph on the same coordinate plane as velocity.
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
In this situation, the velocity is positive/negative and increasing/decreasing?
When velocity is increasing/decreasing, we know that acceleration is positive/negative?
When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing?
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2.2A - Rate of Change

30. Your Turn For which Examples 3 to 6 was the speed increasing? Explain.
When the spend in increasing, the velocity and acceleration have same/opposite signs?
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2.2A - Rate of Change

31. Example 7
The graph below represents the velocity, v(t), in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 11 seconds. It consists of a semicircle and two line segments.
At what time [0, 11], is the speed of the particle the greatest?
At which times, t = 2, t = 6, or t = 9 where the acceleration the greatest? Explain.
Over what time intervals is the particle moving left? Explain.
Over what time intervals is the speed of the particle decreasing? Explain.
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2.2A - Rate of Change

32. Example 7a
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. At what time [0, 11], is the speed of the particle the greatest?
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2.2A - Rate of Change

33. Example 7b
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. At which times, t = 2, t = 6, or t = 9 where the acceleration the greatest? Explain.
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2.2A - Rate of Change

34. Example 7c
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. Over what time intervals is the particle moving left? Explain.
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2.2A - Rate of Change

35. Example 7d
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. Over what time intervals is the speed of the particle decreasing? Explain.
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2.2A - Rate of Change

36. Example 8
The graph below represents the velocity, v(t), in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 11 seconds. It consists of a semicircle and two line segments.
If t = 4, is the particle moving to the right or left? Explain the answer.
Over what time interval is the particle moving to the left? Explain.
At t = 4 seconds, is the acceleration of the particle positive or negative? Explain.
Is there guaranteed to be a time t in the interval, [2, 4] such that v’(t) = –3/2 ft/sec2? Justify answer.
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2.2A - Rate of Change

37. Example 8a
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. If t = 4, is the particle moving to the right or left? Explain the answer.
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2.2A - Rate of Change

38. Example 8b
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. Over what time interval is the particle moving to the left? Explain.
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2.2A - Rate of Change

39. Example 8c
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. At t = 4 seconds, is the acceleration of the particle positive or negative? Explain.
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2.2A - Rate of Change

40. Example 8d
The graph below represents the velocity, v, in feet per second, of a particle moving along the x-axis over the time interval from t = 0 and t = 9 seconds. Is there guaranteed to be a time t in the interval, [2, 4] such that v’(t) = –3/2 ft/sec2? Justify answer.
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2.2A - Rate of Change

41. Example 9
The data below in the table gives the selected values of velocity, in meters/minute, of a particle moving along the x-axis. The velocity v is differentiable function of time, t.
If t = 0, is the particle moving to the right or left? Explain the answer.
Is there a time during the interval [0, 12] minutes when the particle is at rest? Explain answer.
Use the data from the table to approximate v’(10) and explain the meaning of v’(10) in terms of the motion of the particle.
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2.2A - Rate of Change

42. Example 9a
The data below in the table gives the selected values of velocity, in meters/minute, of a particle moving along the x-axis. The velocity v is differentiable function of time, t. If t = 0, is the particle moving to the right or left? Explain the answer.
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2.2A - Rate of Change

43. Example 9b
The data below in the table gives the selected values of velocity, in meters/minute, of a particle moving along the x-axis. The velocity v is differentiable function of time, t. Is there a time during the interval [0, 12] minutes when the particle is at rest? Explain answer.
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2.2A - Rate of Change

44. Example 9c
The data below in the table gives the selected values of velocity, in meters/minute, of a particle moving along the x-axis. The velocity v is differentiable function of time, t. Use the data from the table to approximate v’(10) and explain the meaning of v’(10) in terms of the motion of the particle.
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2.2A - Rate of Change

45. Your Turn
Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 < t < 80 seconds, as shown in the table.
If t = 0, is the particle moving to the right or left? Explain the answer.
Find the average acceleration of Rocket A over the time interval, [0, 80].
Use the data from the table to approximate v’(15) and explain the meaning of v’(15) in terms of the motion of the particle.
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3.7 – Particle Motion