Accumulation and Particle Motion Section 7.5A Calculus AP/Dual, Revised ©2015 viet.dang@humble.k12.tx.us 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

7.5A: Accumulation and Particle Motion Labeling Answers T = Time U = Units N = Noun 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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? 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

How To Input Derivatives into TI-84 Calculator Solve for 𝒅 𝒅𝒙 𝒙𝟐 when 𝒙=𝟓 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

How To Input Integrals into TI-84 Calculator Solve for −𝟐 𝟐 𝒙 𝟐 𝒅𝒙 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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? 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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? 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

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. 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion

7.5A: Accumulation and Particle Motion Assignment Worksheet 12/3/2018 6:40 PM 7.5A: Accumulation and Particle Motion