Section 3.7 Calculus AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us Particle Motion Section 3.7 Calculus AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us 11/29/2018 11:50 AM 3.7 – Particle Motion

What is Position, Velocity, and Acceleration? www.nbclearn.com/nfl/cuecard/50770 11/29/2018 11:50 AM 3.7 – Particle Motion

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) 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

Graph 11/29/2018 11:50 AM 3.7 – Particle Motion

Website http://www.cengage.com/math/discipline_content/stewartcalcet7/2008/14_cengage_tec/publish/deployments/transcendentals_7e/m3_7_sa.swf http://phet.colorado.edu/simulations/sims.php?sim=The_Moving_Man 11/29/2018 11:50 AM 3.7 – Particle Motion

Geometric Sketchpad 11/29/2018 11:50 AM 3.7 – Particle Motion

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. 11/29/2018 11:50 AM 3.7 – Particle Motion

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. 11/29/2018 11:50 AM 3.7 – Particle Motion

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. 11/29/2018 11:50 AM 3.7 – Particle Motion

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. 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

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, ∞) 11/29/2018 11:50 AM 3.7 – Particle Motion

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) (+)(+) 11/29/2018 11:50 AM 3.7 – Particle Motion

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) 11/29/2018 11:50 AM 3.7 – Particle Motion

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) (+)(+) 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

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. 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 3.7 – Particle Motion

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 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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? 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 2.2A - Rate of Change

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. 11/29/2018 11:50 AM 3.7 – Particle Motion