PARTICLE MOTION. Section 3.7A Calculus BC AP/Dual, Revised /30/2018 1:20 AM 3.7A: Particle Motion 1

PARTICLE MOTION Section 3.7A Calculus BC AP/Dual, Revised 2017 viet.dang@humbleisd.net 7/30/2018 1:20 AM 3.7A: Particle Motion 1

WHEN YOU SEE THINK When you see Think Initially t = 0 At rest v t = 0 At the origin x t = 0 Velocity is POSITIVE Particle is moving RIGHT Velocity is NEGATIVE Particle is moving LEFT Average Velocity ( Given x t ) s b s a b a Instantaneous Velocity Velocity at an exact moment POSITIVE acceleration Velocity is increasing NEGATIVE acceleration Velocity is decreasing Instantaneous Speed 7/30/2018 1:20 AM 3.7A: Particle Motion 2 v t

WHAT IS POSITION, VELOCITY, AND ACCELERATION? www.nbclearn.com/nfl/cuecard/50770 7/30/2018 1:20 AM 3.7A: Particle Motion 3

MOTIONS A. Position is s t or x t ; also known as speed is the rate of motion 1. Label could be known as meters 2. Initially means when t = 0 3. At the origin means x t = 0 B. Velocity is v t = s t ; absolute value rate of motion or known as SPEED and DIRECTION 1. Label could be known as meters/second or speed/time 2. At rest means v t = 0 3. If the velocity of the particle is POSITIVE, then the particle is moving to the right 4. If the velocity of the particle is NEGATIVE, then the particle is moving to the left 5. If the order of the particle CHANGES SIGNS, the velocity must change signs 7/30/2018 1:20 AM 3.7A: Particle Motion 4

MOTIONS C. Acceleration known as a t = v t = s t 1. Label could be known as units time 2 2. If the acceleration of the particle is POSITIVE, then the particle is increasing 3. If the acceleration of the particle is NEGATIVE, then the particle is decreasing 4. If a particle Slows Down, signs from v t and s t are different (SIGNS DIFFERENT) 7/30/2018 1:20 AM 3.7A: Particle Motion 5

A. Total Distance = Candidates Test FORMULAS B. Average Velocity = change in time s b s a b a C. Instantaneous Speed = v t or divide the change in position by the Position Velocity Left Right Acceleration Increasing (Above x-axis) Decreasing (Below x-axis) 7/30/2018 1:20 AM 3.7A: Particle Motion 6

GRAPH 7/30/2018 1:20 AM 3.7A: Particle Motion 7

TECHNIQUES OF SPEEDING UP/SLOWING DOWN 7/30/2018 1:20 AM 3.7A: Particle Motion 8

WEBSITES http://www.cengage.com/math/discipline_content/stewartcalcet7/200 8/14_cengage_tec/publish/deployments/transcendentals_7e/m3_7_ sa.swf http://phet.colorado.edu/simulations/sims.php?sim=the_moving_man 7/30/2018 1:20 AM 3.7A: Particle Motion 9

EXAMPLE 1A The position function of a particle moving on a straight line is s t = 2t 3 10t 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. s t = 2t 10t + 5 3 2 3 2 s 1 = 2 1 10 1 + 5 s ( 1) = 3 s 1 = 3 meters 7/30/2018 1:20 AM 3.7A: Particle Motion 10

EXAMPLE 1B The position function of a particle moving on a straight line is s t = 2t 3 10t 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. s t = 2t 10t + 5 3 2 v t = s ' t = 6t 20t 2 v 1 = 6 1 20 1 2 v 1 = 14 m/ s 7/30/2018 1:20 AM 3.7A: Particle Motion 11

EXAMPLE 1C The position function of a particle moving on a straight line is s t = 2t 3 10t 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. v t = 6t 20t 2 a t = v ' t = 12t 20 a ( 1) = 12( 1) 20 a 1 = 8 m/ s 2 7/30/2018 1:20 AM 3.7A: Particle Motion 12

EXAMPLE 1D The position function of a particle moving on a straight line is s t = 2t 3 10t 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. v t v 1 = 6t 20t 2 v 1 = 6 1 20 1 2 v 1 = 14 m / s 7/30/2018 1:20 AM 3.7A: Particle Motion 13

EXAMPLE 2A The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? Velocity at time t x ' t = v t 2 s t = 2t 19t + 12t 7 3 2 v t = 6t 38t + 12 7/30/2018 1:20 AM 3.7A: Particle Motion 14

EXAMPLE 2B The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? Acceleration x '' t = v ' t = a t s t = 2t 19t + 12t 7 v t = 6t 38t + 12 3 2 2 a t = 12t 38 7/30/2018 1:20 AM 3.7A: Particle Motion 15

EXAMPLE 2C The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? v( t ) = 0 2 6t 38t+ 12 = 0 ( 2 t t ) 2 3 19 + 6 = 0 At Rest 2 3t 1 t 6 = 0 1 sec,6secs 3 7/30/2018 1:20 AM 3.7A: Particle Motion 16 t =

EXAMPLE 2D The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? ( t )( t ) 2 3 1 6 = 0 Moving furthest to the Left (Relative MINIMUM) 2 v t = 6t 38t + 12 1 t =, t = 6 3 x t = 0 f 0 0, 7 ( t) v( t) changes signs from NEG to POS s is moving furthest to the left at point 6,187 when 7/30/2018 1:20 AM 3.7A: Particle Motion 17 0, 1 3 1 f 4 POSITIVE RIGHT x t = 1 3 f 1 3 1 3, 136 27 Rel MAX 1 x 3, 6 t = 6 (6, ) f 2 + NEGATIVE LEFT f(6) 6, 187 Rel MIN f 7 + + POSITIVE RIGHT

EXAMPLE 2E The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? The particle is moving to the right at 1 I 0, ( 6, ) when v( t) is where f ' >0 3 Moving to the RIGHT x = 0 f 0 0, 7 0, 1 3 1 f 4 POSITIVE RIGHT 7/30/2018 1:20 AM 3.7A: Particle Motion 18 x = 1 3 f 1 3 1 3, 136 27 Rel MAX 1 3, 6 x = 6 (6, ) f 2 + NEGATIVE LEFT f(6) 6, 187 Rel MIN f 7 + + POSITIVE RIGHT

EXAMPLE 2F The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) slowing down? SLOWING DOWN (get critical values and POI) v ' = a t = 12t 38 = 0 t = 19 6 7/30/2018 1:20 AM 3.7A: Particle Motion 19

EXAMPLE 2G The position function of a particle moving on a straight line is s t = 2t 3 19t 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 to the left, e) moving to the right, and f) v t a t = = slowing down? 1,6 3 19 6 x = 0 f 0 0, 7 x = 0 f(0) 0, 7 f x = 12t 38 7/30/2018 1:20 AM 3.7A: Particle Motion 20 0, 1 3 f 1 4 ( )( ) POSITIVE, RIGHT x = 1/3 0, 19 6 f 1f 1 3 1 3, 136 27 NEGATIVE Rel MAX 1x 3, 6 = 19/6 x = 6 19 (6, ) 6, f 2 + f 19 NEGATIVE 6 LEFT + + + + + + + + + + + + + v( t ) 1 0 3 6 ++++++++++++++++ a( t ) 1 19 0 0,,6 3 6 19 6 3.1667 f(6) 6, 187 Rel MIN f 7 f (4) + + (+) POSITIVE RIGHT POSITIVE

EXAMPLE 2G 7/30/2018 1:20 AM 3.7A: Particle Motion 21

YOUR TURN The position function of a particle moving on a straight line is x t = 3t 4 16t 3 + 24t 2 from 0, 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 slows down, e) identify the velocity when acceleration is first zero 7/30/2018 1:20 AM 3.7A: Particle Motion 22

SPEED, VELOCITY, AND TANGENT LINES A. Speed is the absolute value of velocity. It is measured of how fast something is moving with the regard of direction B. The effect of how an absolute value function has it on the graph is that it reflects all values that are below and above on the x-axis 7/30/2018 1:20 AM 3.7A: Particle Motion 24

EXAMPLE 3 A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. P(t) graph given. A) Over what time interval is the particle moving to the left? Explain. B) When is the first time that P reverses direction? C) When does P move at its greatest speed? Explain. D) Is there guaranteed to be a time t in the interval, 0, 3 such that v t = 3 ft? Justify 8 sec answer. 7/30/2018 1:20 AM 3.7A: Particle Motion 25

EXAMPLE 3A A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. Over what time interval is the particle moving to the left? Explain. ) ( s The particle is moving to the left from 0,1 and 4,6 because ' t 0 7/30/2018 1:20 AM 3.7A: Particle Motion 26

EXAMPLE 3B A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. When is the first time that P reverses direction? P reverses direction at t = 1 because direction changes from left to right. 7/30/2018 1:20 AM 3.7A: Particle Motion 27

EXAMPLE 3C A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. When does P move at its greatest speed? Explain. The particle is at its greatest speed from ( 4,6 because speed is the absolute value of velocity 7/30/2018 1:20 AM 3.7A: Particle Motion 28

EXAMPLE 3D A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. Is there guaranteed to be a time t in the interval, 0, 3 such that v t = 3 8 Justify answer. MVT does not apply between 0 and 3 because v( t) is not differentiable. Continuity does not validate differentiability. ft sec? 7/30/2018 1:20 AM 3.7A: Particle Motion 29

EXAMPLE 4 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 & t = 11 seconds. It consists of a semicircle and two line segments. A) At what time 0, 11, is the speed of the particle the greatest? B) At which times, t = 2, t = 6, or t = 9 where the acceleration the greatest? Explain. C) Over what time intervals is the particle moving left? Explain. D) Over what time intervals is the speed of the particle decreasing? Explain. 7/30/2018 1:20 AM 3.7A: Particle Motion 30

EXAMPLE 4A 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 & 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 t=8 when speed is 6 feet/sec 7/30/2018 1:20 AM 3.7A: Particle Motion 31

EXAMPLE 4B 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 & 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. At t = 9 when acceleration is at 3 seconds whereas 3 when t = 2 slope is m = 0 and where t = 6, slope is. 2 7/30/2018 1:20 AM 3.7A: Particle Motion 32

EXAMPLE 4C 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 & t = 11 seconds. It consists of a semicircle and two line segments. Over what time intervals is the particle moving left? Explain. 4,10 because vt < 0 7/30/2018 1:20 AM 3.7A: Particle Motion 33

EXAMPLE 4D 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 & t = 11 seconds. It consists of a semicircle and two line segments. Over what time intervals is the speed of the particle decreasing? Explain. At ( 2, 4 ) because v( t) > 0 and a ( t) < 0 and at ( 8,10 ) because v ( t ) < 0 and a ( t) > 0 when the signs different of v( t) and a ( t ), the particle slows down. 7/30/2018 1:20 AM 3.7A: Particle Motion 34

YOUR TURN A particle P (position) moves along the x-axis over the time interval from t = 0 and t = 6 seconds where x stands for seconds and y is in feet. This graph shows velocity. A) Over what time interval is the particle moving to the left? Explain. At 1,4 because v( t) < 0 B) Over what time intervals is the speed of the particle decreasing? Explain. ) v t a ( t) At 0,1 because v t > 0 and a t <0 and at 3,4 because < 0 and >0 When the signs change of v( t) and a( t), the particle slows down. C) Is there guaranteed to be a time t in the interval, 0, 2 where the particle is at rest? Explain. Yes, between t = 0 and t = 2 where the velocity must be 0. 7/30/2018 1:20 AM 3.7A: Particle Motion 35 Therefore, the IVT must exist and differentiability implies continuity

EXAMPLE 5 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. Time t (min) Velocity v(t) (meters/min) 0 2 5 6 8 12 3 2 3 5 7 5 A) If t = 0, is the particle moving to the right or left? Explain the answer. B) Is there a time during the interval 0, 12 minutes when the particle is at rest? Explain answer. C) Use the data to the table to approximate v 10 and explain the meaning of v 10 in terms of the motion of the particle. 7/30/2018 1:20 AM 3.7A: Particle Motion 36

EXAMPLE 5A 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. Time t (min) Velocity v(t) (meters/min) 0 2 5 6 8 12 3 2 3 5 7 5 If t = 0, is the particle moving to the right or left? Explain the answer. Left because v( t) 0 7/30/2018 1:20 AM 3.7A: Particle Motion 37

EXAMPLE 5B 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. Time t (min) 0 2 5 6 8 12 Velocity v(t) (meters/min) 3 2 3 5 7 5 Is there a time during the interval 0, 12 minutes when the particle is at rest? Explain answer. Yes, between t = 0 and t = 2 where the velocity must be 0. Therefore, the IVT must exist and differentiability implies continuity 7/30/2018 1:20 AM 3.7A: Particle Motion 38

EXAMPLE 5C 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. Time t (min) Velocity v(t) (meters/min) Use the data to the table to approximate v (10) and explain the meaning of v (10) in terms of the motion of the particle. s( 12) s( 8) 5 7 2 2 v' ( t) meters / min 12 8 12 8 4 v' 10 is the instantaneous acceleration of the particle when t=10 min 0 2 5 6 8 12 3 2 3 5 7 5 7/30/2018 1:20 AM 3.7A: Particle Motion 39

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. t (secs) 0 10 20 30 40 50 60 70 80 v(t) (ft/sec. ) 5 14 22 29 35 40 44 47 49 A) Find the average acceleration of rocket A over the interval 0, 80 seconds. Indicate the units of measurement. B) Is there a time during the interval 0, 12 minutes when the particle is at rest? Explain answer. C) Use the data to the table to approximate v 45 and explain the meaning of v 45 in terms of the motion of the particle. 7/30/2018 1:20 AM 3.7A: Particle Motion 40

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) The position of a particle moving along the x-axis is x t = sin 2t cos 3t for time t 0. When t = π, the acceleration of the particle is: Vocabulary Connections and Process Answer and Justifications Velocity Acceleration = sin ( 2 ) cos( 3 ) ' = = 2 cos( 2 ) + 3sin ( 3 ) '' = ' = = 4sin ( 2 ) + 9cos( 3 ) x t t t x t v t t t x t v t a t t t C Chain Rule x ''( ) = v '( ) = a ( ) = 4sin ( 2( )) + 9cos( 3( )) a a ( ) = 4( 0) + 9( 1) ( ) = 9 7/30/2018 1:20 AM 3.7A: Particle Motion 43

ASSIGNMENT Worksheet 7/30/2018 1:20 AM 3.7A: Particle Motion 44