PARTICLE MOTION: DAY 2
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1 PARTICLE MOTION: DAY 2 Section 3.6A Calculus AP/Dual, Revised 2018 viet.dang@humbleisd.net 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 1
2 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 1:24 AM 3.6A: Particle Motion Day 2 2 v t
3 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 1:24 AM 3.6A: Particle Motion Day 2 3
4 IMAGINE Stepping on the gas, Acceleration pulls into a direction that you are moving. Does your speed increase or decrease? Going too fast is not good so you put on your breaks. Now, you accelerate in the opposite direction but you are moving forward. Does your speed increase or decrease? Finally, you stop and shift into reverse and start moving backwards (negative velocity) and then you push the on the gas to accelerate in the negative direction. Does your speed increase or decrease? Then, you pull on the breaks after all of that. Speed increases or decreases? 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 4
5 EXAMPLE 1 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 Speed /30/2018 1:24 AM 3.6A: Particle Motion Day 2 5
6 EXAMPLE 1 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. A) In this situation, the velocity is positive/negative and increasing/decreasing? B) When velocity is increasing/decreasing, we know that acceleration is positive/negative? C) When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing? 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 6
7 EXAMPLE 2 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 Speed /30/2018 1:24 AM 3.6A: Particle Motion Day 2 7
8 EXAMPLE 2 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. A) In this situation, the velocity is positive/negative and increasing/decreasing? B) When velocity is increasing/decreasing, we know that acceleration is positive/negative? C) When examining the graph of speed and table of values, the conclusion is that speed is increasing/decreasing? 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 8
9 EXAMPLE 3 Given this graph below of v t, fill out the table below for intervals 0, a and c, d Interval 0, a [c, d] Velocity Positive or Negative Acceleration Positive or Negative Speed Increasing or Decreasing Inc. + Dec. 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 9
10 YOUR TURN Given this graph below of v t, fill out the table below. Then, identify the values of t at which the speed obtains its local and absolute maximum(s). When do, the minimum speeds occur? When are they? Local Maxs: t = c, t = e Absolute Max: t = a Absolute Min of 0: t = 0, t = b, t = d, t = f Interval a, b b, c Velocity Positive or Negative Acceleration Positive or Negative Speed Increasing or Decreasing + Dec. + + Inc. 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 10
11 EXAMPLE 4 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 1:24 AM 3.6A: Particle Motion Day 2 11
12 EXAMPLE 4A 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 1:24 AM 3.6A: Particle Motion Day 2 12
13 EXAMPLE 4B 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 1:24 AM 3.6A: Particle Motion Day 2 13
14 EXAMPLE 4C 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 1:24 AM 3.6A: Particle Motion Day 2 14
15 EXAMPLE 4D 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 1:24 AM 3.6A: Particle Motion Day 2 15
16 EXAMPLE 5 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 1:24 AM 3.6A: Particle Motion Day 2 16
17 EXAMPLE 5A 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 1:24 AM 3.6A: Particle Motion Day 2 17
18 EXAMPLE 5B 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 1:24 AM 3.6A: Particle Motion Day 2 18
19 EXAMPLE 5C 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 1:24 AM 3.6A: Particle Motion Day 2 19
20 EXAMPLE 5D 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 1:24 AM 3.6A: Particle Motion Day 2 20
21 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 1:24 AM 3.6A: Particle Motion Day 2 21 Therefore, the IVT must exist and differentiability implies continuity
22 EXAMPLE 6 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) 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 1:24 AM 3.6A: Particle Motion Day 2 22
23 EXAMPLE 6A 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) If t = 0, is the particle moving to the right or left? Explain the answer. Left because v( t) 0 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 23
24 EXAMPLE 6B 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) 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 1:24 AM 3.6A: Particle Motion Day 2 24
25 EXAMPLE 6C 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) v' ( t) meters / min v' 10 is the instantaneous acceleration of the particle when t=10 min ( ) /30/2018 1:24 AM 3.6A: Particle Motion Day 2 25
26 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) v(t) (ft/sec. ) 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 seconds 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 1:24 AM 3.6A: Particle Motion Day 2 26
27 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) v(t) (ft/sec. ) 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 seconds 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 A) ft / sec 20 B) No because 0 never falls below zero. ( ) v( ) ( ) v( ) v Cv ) '( 45 ) feet / sec v' 45 is the Instantaneous acceleration of the particle when t=45 secs 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 27
28 ASSIGNMENT Worksheet 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 28
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