1. Accumulation and Particle Motion
Section 7.5A
Calculus AP/Dual, Revised ©2017
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7.5A: Accumulation and Particle Motion

2. 7.5A: Accumulation and Particle Motion
Integral of Motion
𝒗 𝒕 𝒅𝒕 is an INDEFINITE integral.
It will given an expression for position at time, 𝒕.
Do not forget the +𝑪, the value of which it can be determined if one knows about position value at a particular time.
DISPLACEMENT: 𝒕 𝟏 𝒕 𝟐 𝒗 𝒕 𝒅𝒕 is a DEFINITE integral and so the answer will be number.
By the Fundamental Theorem of Calculus, the integral will yield 𝒙 𝒕 𝟐 −𝒙 𝒕 𝟏 since 𝒗 𝒕 = 𝒙 ′ 𝒕 . It is also known as DISPLACEMENT where the answer could be known as positive or negative depending upon the particle landing to the right or left of its original position
TOTAL DISTANCE: 𝒕 𝟏 𝒕 𝟐 𝒗 𝒕 𝒅𝒕 is a DEFINITE integral and so the answer will be number.
The answer will represent the total distance traveled by the particle over time
The answer should always be non-negative
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7.5A: Accumulation and Particle Motion

3. 7.5A: Accumulation and Particle Motion
Labeling Answers
T = Time U = Units N = Noun
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7.5A: Accumulation and Particle Motion

4. 7.5A: Accumulation and Particle Motion
Review
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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7.5A: Accumulation and Particle Motion

5. 7.5A: Accumulation and Particle Motion
Review 1a
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?
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7.5A: Accumulation and Particle Motion

6. 7.5A: Accumulation and Particle Motion
Review 1b
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 which times, 𝒕=𝟐, 𝒕=𝟔, or 𝒕=𝟗 where the acceleration the greatest? Explain.
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7.5A: Accumulation and Particle Motion

7. 7.5A: Accumulation and Particle Motion
Review 1c
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. Over what time intervals is the particle moving left? Explain.
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7.5A: Accumulation and Particle Motion

8. 7.5A: Accumulation and Particle Motion
Review 1d
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. Over what time intervals is the speed of the particle decreasing? Explain.
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7.5A: Accumulation and Particle Motion

9. 7.5A: Accumulation and Particle Motion
Example 1
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.
Find the total distance traveled by the particle over the time interval, 𝟎<𝒕<𝟏𝟏.
Find the value of 𝟎 𝟏𝟏 𝒗 𝒕 𝒅𝒕 and explain the meaning of this integral in the context of the problem.
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7.5A: Accumulation and Particle Motion

10. 7.5A: Accumulation and Particle Motion
Example 1a
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.
Find the total distance traveled by the particle over the time interval, 𝟎<𝒕<𝟏𝟏.
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7.5A: Accumulation and Particle Motion

11. 7.5A: Accumulation and Particle Motion
Example 1b
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.
B) Find the value of 𝟎 𝟏𝟏 𝒗 𝒕 𝒅𝒕 and explain the meaning of this integral in the context of the problem.
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7.5A: Accumulation and Particle Motion

12. How To Input Derivatives into TI-84 Calculator
Solve for 𝒅 𝒅𝒙 𝒙𝟐 when 𝒙=𝟓
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7.5A: Motion

13. How To Input Integrals into TI-84 Calculator
Solve for −𝟐 𝟐 𝒙 𝟐 𝒅𝒙
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7.5A: Accumulation and Particle Motion

14. 7.5A: Accumulation and Particle Motion
Example 2
The rate of change, in kilometers per hour, of the altitude of a hot air balloon is given by 𝒓 𝒕 = 𝒕 𝟑 −𝟒 𝒕 𝟐 +𝟔 for time 𝟎≤𝒕≤𝟒, where 𝒕 is measured in hours. Assume the balloon is initially at ground level. Use the calculator to solve.
For what values of 𝒕, 𝟎≤𝒕≤𝟒, is the altitude of the balloon decreasing?
Find the value of 𝒓 ′ 𝟐 and explain the meaning of the answer in the context of the problem.
Find the value of 𝟎 𝟒 𝒓 𝒕 𝒅𝒕 and explain the meaning of the answer in the context of the problem. Indicate the units of measurement.
Find the value of 𝟎 𝟒 𝒓 𝒕 𝒅𝒕 and explain the meaning of the answer in the context of the problem. Indicate the units of measurement.
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7.5A: Accumulation and Particle Motion

15. 7.5A: Accumulation and Particle Motion
Example 2a
The rate of change, in kilometers per hour, of the altitude of a hot air balloon is given by 𝒓 𝒕 = 𝒕 𝟑 −𝟒 𝒕 𝟐 +𝟔 for time 𝟎≤𝒕≤𝟒, where 𝒕 is measured in hours. Assume the balloon is initially at ground level. Use the calculator to solve.
For what values of 𝒕, 𝟎≤𝒕≤𝟒, is the altitude of the balloon decreasing?
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7.5A: Accumulation and Particle Motion

16. 7.5A: Accumulation and Particle Motion
Example 2b
The rate of change, in kilometers per hour, of the altitude of a hot air balloon is given by 𝒓 𝒕 = 𝒕 𝟑 −𝟒 𝒕 𝟐 +𝟔 for time 𝟎≤𝒕≤𝟒, where 𝒕 is measured in hours. Assume the balloon is initially at ground level. Use the calculator to solve. B) Find the value of 𝒓 ′ 𝟐 and explain the meaning of the answer in the context of the problem.
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7.5A: Accumulation and Particle Motion

17. 7.5A: Accumulation and Particle Motion
Example 2c
The rate of change, in kilometers per hour, of the altitude of a hot air balloon is given by 𝒓 𝒕 = 𝒕 𝟑 −𝟒 𝒕 𝟐 +𝟔 for time 𝟎≤𝒕≤𝟒, where 𝒕 is measured in hours. Assume the balloon is initially at ground level. Use the calculator to solve. C) Find the value of 𝟎 𝟒 𝒓 𝒕 𝒅𝒕 and explain the meaning of the answer in the context of the problem. Indicate the units of measurement.
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7.5A: Accumulation and Particle Motion

18. 7.5A: Accumulation and Particle Motion
Example 2d
The rate of change, in kilometers per hour, of the altitude of a hot air balloon is given by 𝒓 𝒕 = 𝒕 𝟑 −𝟒 𝒕 𝟐 +𝟔 for time 𝟎≤𝒕≤𝟒, where 𝒕 is measured in hours. Assume the balloon is initially at ground level. Use the calculator to solve. D) Find the value of 𝟎 𝟒 𝒓 𝒕 𝒅𝒕 and explain the meaning of the answer in the context of the problem. Indicate the units of measurement.
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7.5A: Accumulation and Particle Motion

19. 7.5A: Accumulation and Particle Motion
Example 3
An object traveling in a straight line has position 𝒙 𝒕 at time 𝒕. If the initial position is 𝒙 𝟎 =𝟐 and the velocity of the object is 𝟑 𝟏+ 𝒕 𝟐 , what is the position of the object at time 𝒕=𝟑?
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7.5A: Accumulation and Particle Motion

20. 7.5A: Accumulation and Particle Motion
Your Turn
The velocity of a particle moving along the 𝒙-axis is given by 𝒗 𝒕 = 𝒕 𝟏/𝟐 𝐜𝐨𝐬⁡𝒕 for time 𝟎≤𝒕≤𝟔.𝟓, where 𝒕 is measured in hours. When 𝒕=𝟎, the particle is at 𝒙=−𝟏. Use the calculator to solve.
Write an expression for the position of the particle at any time 𝒕
Determine when the particle is farthest from the origin
When the particle is farthest from the origin, is the velocity increasing or decreasing. Explain.
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7.5A: Accumulation and Particle Motion

21. 7.5A: Accumulation and Particle Motion
Your Turn
The velocity of a particle moving along the 𝒙-axis is given by 𝒗 𝒕 = 𝒕 𝟏/𝟐 𝐜𝐨𝐬⁡𝒕 for time 𝟎≤𝒕≤𝟔.𝟓, where 𝒕 is measured in hours. When 𝒕=𝟎, the particle is at 𝒙=−𝟏. Use the calculator to solve.
Write an expression for the position of the particle at any time 𝒕
Determine when the particle is farthest from the origin 𝟎
𝝅 𝟐
𝟑𝝅 𝟐
𝟔.𝟓 −𝟏
−𝟎.𝟐𝟗𝟓𝟗
−𝟑.𝟖𝟐𝟗𝟏
−𝟎.𝟖𝟖𝟔𝟓
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7.5A: Accumulation and Particle Motion

22. 7.5A: Accumulation and Particle Motion
Your Turn
The velocity of a particle moving along the 𝒙-axis is given by 𝒗 𝒕 = 𝒕 𝟏/𝟐 𝐜𝐨𝐬⁡𝒕 for time 𝟎≤𝒕≤𝟔.𝟓, where 𝒕 is measured in hours. When 𝒕=𝟎, the particle is at 𝒙=−𝟏. Use the calculator to solve.
C) When the particle is farthest from the origin, is the velocity increasing or decreasing. Explain.
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7.5A: Accumulation and Particle Motion

23. 7.5A: Accumulation and Particle Motion
Assignment
Worksheet
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7.5A: Accumulation and Particle Motion