Chapter 2 - Representing Motion w./ QuickCheck Questions
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1 Chapter 2 - Representing Motion w./ QuickCheck Questions 2015 Pearson Education, Inc. Anastasia Ierides Department of Physics and Astronomy University of New Mexico August 27, 2015
2 Review of Last Time Looking at motion diagrams, how do we represent position & velocity on position & velocity graphs? Looking at position graphs, how do we represent the velocity of object on a velocity graph? & vice-versa How do we find the instantaneous velocity? the acceleration?
3 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Which position-versus-time graph matches this motion diagram?
4 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Same displacement starting from x < 0 Which position-versus-time graph matches this motion diagram?
5 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Same displacement starting from x < 0 Which position-versus-time graph matches this motion diagram?
6 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Same displacement starting from x < 0 Which position-versus-time graph matches this motion diagram?
7 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Larger displacement at and after from x = 0 Same displacement starting from x < 0 Which position-versus-time graph matches this motion diagram?
8 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Same displacement at and after from x = 0 Which position-versus-time graph matches this motion diagram?
9 QuickCheck Question 2.1 Here is a motion diagram of a car moving along a straight road: Same displacement at and after from x = 0 Which position-versus-time graph matches this motion diagram? E.
10 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? E.None of the above.
11 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Small velocity vectors w./ equal length moving to the right, from x < 0 Which velocity-versus-time graph matches this motion diagram? E.None of the above.
12 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Small velocity vectors w./ equal length moving to the right, from x < 0 Which velocity-versus-time graph matches this motion diagram? E.None of the above.
13 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Small, nonzero velocity vectors moving to the right, vx 0 Which velocity-versus-time graph matches this motion diagram? E.None of the above.
14 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Small, nonzero velocity vectors moving to the right, vx 0 Which velocity-versus-time graph matches this motion diagram? E.None of the above.
15 QuickCheck Question 2.2 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? E.None of the above. C.
16 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram?
17 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Nonzero velocity vectors moving to the left, vx 0 Which velocity-versus-time graph matches this motion diagram?
18 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Nonzero velocity vectors moving to the left throughout, vx 0 Which velocity-versus-time graph matches this motion diagram?
19 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Velocity vectors are small first, then large Which velocity-versus-time graph matches this motion diagram?
20 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Velocity vectors are small first, then large Which velocity-versus-time graph matches this motion diagram?
21 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Velocity vectors are small first, then large Which velocity-versus-time graph matches this motion diagram?
22 QuickCheck Question 2.3 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? D.
23 QuickCheck Question 2.5 The slope at a point on a position-versus-time graph of an object is A. The object s speed at that point B. The object s velocity at that point. C. The object s acceleration at that point. D. The distance traveled by the object to that point. E. I am not sure.
24 QuickCheck Question 2.5 The slope at a point on a position-versus-time graph of an object is A. The object s speed at that point B. The object s velocity at that point. C. The object s acceleration at that point. D. The distance traveled by the object to that point. E. I am not sure.
25 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: Which of the following velocity graphs matches the position graph? A. B. C. D.
26 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: Here we see that the object is in the same position for some t The slope is 0 meaning that v = 0 Which of the following velocity graphs matches the position graph? A. B. C. D.
27 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: Here we see that the object is in the same position for some t The slope is 0 meaning that v = 0 Which of the following velocity graphs matches the position graph? A. B. C. D.
28 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: The slope is negative Here we see that the and constant object is moving toward meaning that v < 0 the negative direction and constant Which of the following velocity graphs matches the position graph? A. B. C. D.
29 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: The slope is negative Here we see that the and constant object is moving toward meaning that v < 0 the negative direction Which of the following velocity graphs matches the position graph? A. B. C. D.
30 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: The slope is negative Here we see that the and constant object is moving toward meaning that v < 0 the negative direction and constant Which of the following velocity graphs matches the position graph? A. B. C. D.
31 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: The slope is negative Here we see that the and constant object is moving toward meaning that v < 0 the negative direction and constant Which of the following velocity graphs matches the position graph? A. B. C. D.
32 QuickCheck Question 2.4 A graph of position versus time for a basketball player moving down the court appears as follows: Which of the following velocity graphs matches the position graph? A. B. C. D.
33 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: Which of the following position graphs matches the velocity graph? A. B. C. D.
34 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: Here we see v > 0 and constant Which of the following position graphs matches the velocity graph? A. B. C. D.
35 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: Here we see v > 0 and constant Which of the following position graphs matches the velocity graph? A. B. C. D.
36 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: Here we see v = 0 Which of the following position graphs matches the velocity graph? A. B. C. D.
37 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: There is no motion Which of the following position graphs matches the velocity graph? A. B. C. D.
38 QuickCheck Question 2.6 A graph of velocity versus time for a hockey puck shot into a goal appears as follows: Which of the following position graphs matches the velocity graph? A. B. C. D.
39 QuickCheck Question 2.7 Which velocity-versus-time graph goes with this position graph?
40 QuickCheck Question 2.7 Which velocity-versus-time graph goes with this position graph? Here we have a pretty linear position graph with positive slope
41 QuickCheck Question 2.7 Which velocity-versus-time graph goes with this position graph? which means constant and positive velocity
42 QuickCheck Question 2.7 Which velocity-versus-time graph goes with this position graph? C.
43 QuickCheck Question 2.8 Here is a position graph of an object: At t = 1.5 s, the object s velocity is A.40 m/s B.20 m/s C.10 m/s D. 10 m/s E. None of the above
44 QuickCheck Question 2.8 Here is a position graph of an object: At t = 1.5 s, the object s velocity is A.40 m/s B.20 m/s C.10 m/s D. 10 m/s Let s look at the slope of the line passing through the point E. None of the above
45 QuickCheck Question 2.8 Here is a position graph of an object: At t = 1.5 s, the object s velocity is A.40 m/s B.20 m/s C.10 m/s D. 10 m/s E. None of the above vx = (xf - xi)/(tf - ti) = (20 m - 0 m)/(2 s - 1 s) = 20 m/s
46 QuickCheck Question 2.8 Here is a position graph of an object: At t = 1.5 s, the object s velocity is A.40 m/s B.20 m/s C.10 m/s D. 10 m/s E. None of the above vx = (xf - xi)/(tf - ti) = (20 m - 0 m)/(2 s - 1 s) = 20 m/s
47 Displacement from Velocity How can we find the distance that an object traveled or the displacement that the object experienced from a velocity graph?
48 Displacement from Velocity How can we find the distance that an object traveled or the displacement that the object experienced from a velocity graph? We look at the area under the curve
49 QuickCheck Question 2.11 Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s. At t = 4.0 s, the object s position is A.20 m B.16 m C.12 m D.8 m E. 4 m
50 QuickCheck Question 2.11 Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s. At t = 4.0 s, the object s position is A.20 m B.16 m C.12 m D.8 m E. 4 m
51 QuickCheck Question 2.11 Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s. At t = 4.0 s, the object s position is A.20 m B.16 m C.12 m D.8 m E. 4 m A = (4 m/s)(2 s - 0 s) + 1/2 (4 m/s)(4 s - 2 s) = 8 m + 1/2 (8 m) = + 12 m
52 Instantaneous Velocity If we don t have uniform motion, i.e., constant displacement over each time interval, how do we the object s velocity at each instant of time?
53 Instantaneous Velocity If we don t have uniform motion, i.e., constant displacement over each time interval, how do we the object s velocity at each instant of time? Look at the slope of the tangent to the curve
54 QuickCheck Question 2.9 When do objects 1 and 2 have the same velocity? A. At some instant before time t 0 B. At time t 0 C. At some instant after time t 0 D. Both A and B E. Never
55 QuickCheck Question 2.9 When do objects 1 and 2 have the same velocity? A. At some instant before time t 0 B. At time t 0 C. At some instant after time t 0 D. Both A and B E. Never Same slope at this time
56 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
57 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Let s look at the slope at each point A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
58 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Let s look at the slope at each point A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
59 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Let s look at the slope at each point A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
60 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Let s look at the slope at each point A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
61 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Which line looks parallel to the position graph of P? A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
62 QuickCheck Question 2.10 Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? Which line looks parallel to the position graph of P? A. P and Q have the same velocity at 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.
63 Acceleration What is acceleration? Is it always constant? How do we represent it? In motion diagrams and graphs How do we know if it is positive or negative?
64 Looking back at velocity Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s. What is represented by the slope of this graph? Do we know what Δv x /Δt is? It is the rate of change of velocity.
65 Looking back at velocity In this case we see that the acceleration, a x =Δv x /Δt, is 0 m/s 2 for the first 2 s, and a x = (0 m/s-4 m/s)/(4 s - 2 s) = - 2 m/s 2 Meaning that the object is slowing down
66 Representing Acceleration a x (m/s 2 ) In this case we see that the acceleration, a x =Δv x /Δt, is 0 m/s 2 for the first 2 s, and a x = (0 m/s-4 m/s)/(4 s - 2 s) = - 2 m/s 2 Meaning that the object is slowing down
67 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2?
68 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2? Let s add vectors
69 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2? Let s add vectors
70 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2? Let s add vectors
71 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2? Let s add vectors
72 QuickCheck Question 2.12 A particle has velocity v1 as it moves from point 1 to point 2. The acceleration is shown. What is its velocity vector v2 as it moves away from point 2? Let s add vectors B.
73 QuickCheck Question 2.14 The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
74 QuickCheck Question 2.14 What do we see for the position? Along the axis x is? The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
75 QuickCheck Question 2.14 What do we see for the position? Along the axis x is? The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
76 QuickCheck Question 2.14 What do we see for the velocity? It points in what direction? The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
77 QuickCheck Question 2.14 The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
78 QuickCheck Question 2.14 What do we see for the acceleration? The motion diagram shows a particle that is slowing down. The sign of the position x and the sign of the velocity v x are: A. Position is positive, velocity is positive. B. Position is positive, velocity is negative. C. Position is negative, velocity is positive. D. Position is negative, velocity is negative.
79 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Add up the vectors!
80 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Add up the vectors!
81 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Positive
82 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Add up the vectors!
83 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Positive Negative
84 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Add up the vectors!
85 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Positive Negative Positive
86 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Add up the vectors!
87 QuickCheck Question 2.17 These four motion diagrams show the motion of a particle along the x-axis. 1. Which motion diagrams correspond to a positive acceleration? 2. Which motion diagrams correspond to a negative acceleration? Positive Negative Positive Negative
88 Acceleration Graphs How do we represent acceleration on a graph? If we know what its velocity graph looks like, how can we tell the object s acceleration?
89 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like?
90 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? If it s moving to the right we know that the velocity is?
91 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? If it s moving to the right we know that the velocity is? positive
92 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? If it s moving to the right we know that the velocity is? positive If it s speeding up we know that the velocity is?
93 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? If it s moving to the right we know that the velocity is? positive If it s speeding up we know that the velocity is? increasing
94 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? If it s moving to the right we know that the velocity is? positive If it s speeding up we know that the velocity is? increasing
95 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? Looking at the graphs, we see that the velocity has a constant and positive slope. What does that tells us about the acceleration?
96 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? Looking at the graphs, we see that the velocity has a constant slope. What does that tells us about the acceleration? it s constant
97 QuickCheck Question 2.21 A cart speeds up while moving away from the origin. What do the velocity and acceleration graphs look like? Looking at the graphs, we see that the velocity has a constant and positive slope. What does that tells us about the acceleration? it s constant
98 Acceleration Graphs Can we draw a velocity graph given an acceleration graph?
99 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph?
100 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? Let s look at the first time interval
101 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? Let s look at the first time interval
102 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is negative and constant. What does this tell us?
103 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is negative and constant. What does this tell us? negative slope of velocity
104 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is negative and constant. What does this tell us? negative slope of velocity
105 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? Now let s look at the second time interval
106 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? Now let s look at the second time interval
107 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is positive and constant. What does this tell us?
108 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is positive and constant. What does this tell us? the velocity has a positive slope
109 QuickCheck Question 2.25 Which velocity-versus-time graph goes with this acceleration graph? The acceleration is positive and constant. What does this tell us? the velocity has a positive slope
110 Equations of Motion Consider an object with the following velocity graph
111 Equations of Motion Consider an object with the following velocity graph We can see that the object moves with constant acceleration, ax
112 Equations of Motion Consider an object with the following velocity graph We can see that the object moves with constant acceleration, ax We can easily find the displacement of the object for a time interval, by taking the area under the curve
113 Equations of Motion Consider an object with the following velocity graph But how do we find the final velocity if all we are given is ax, v x,i and t?
114 Equations of Motion Consider an object with the following velocity graph But how do we find the final velocity if all we are given is ax, v x,i and t? We need an equation of motion: v x,f = v x,i + ax t
115 Equations of Motion Consider an object with the following velocity graph Now how do we find the final position if all we are given is ax, v x,i and t?
116 Equations of Motion Consider an object with the following velocity graph Now how do we find the final position if all we are given is ax, v x,i and t? Look at the different areas under the curve: x = v x,i t
117 Equations of Motion Consider an object with the following velocity graph Now how do we find the final position if all we are given is ax, v x,i and t? Look at the different areas under the curve: x = v x,i t + 1 / 2 ax t 2
118 Equations of Motion Consider an object with the following velocity graph Now how do we find the final position if all we are given is ax, v x,i and t? Look at the different areas under the curve: x = v x,i t + 1 / 2 ax t 2 x f = x i + v x,i t + 1 / 2 ax t 2
119 Equations of Motion Consider an object with the following velocity graph Now how do we find the final velocity if all we are given is ax, v x,i and x?
120 Equations of Motion Consider an object with the following velocity graph Now how do we find the final velocity if all we are given is ax, v x,i and x? Combine the previous two equations to cancel out t: (v x,f ) 2 = (v x,i ) ax x
121 Equations of Motion for Constant Acceleration Consider an object with the following velocity graph v x,f = v x,i + ax t x = v x,i t + 1 / 2 ax t 2 x f = x i + v x,i t + 1 / 2 ax t 2 (v x,f ) 2 = (v x,i ) ax x
122 Problem Solving Prepare: visualize it, draw a picture of what s going on; put together all the givens and unknowns or any equations needed; define symbols, units, etc. Solve: do the math, calculations, etc. Assess: does your answer make sense? are the units right?
123 Example 2.12 Calculating the minimum length of a runway A fully loaded Boeing 747 with all engines at full thrust accelerates at 2.6 m/s 2. Its minimum takeoff speed is 70 m/s. How much time will the plane take to reach its takeoff speed? What minimum length of runway does the plane require for takeoff?
124 Example 2.12 Calculating the minimum length of a runway A fully loaded Boeing 747 with all engines at full thrust accelerates at 2.6 m/s 2. Its minimum takeoff speed is 70 m/s. How much time will the plane take to reach its takeoff speed? What minimum length of runway does the plane require for takeoff? PREPARE The visual overview of FIGURE 2.33 summarizes the important details of the problem. We set x i and t i equal to zero at the starting point of the motion, when the plane is at rest and the acceleration begins. The final point of the motion is when the plane achieves the necessary takeoff speed of 70 m/s. The plane is accelerating to the right, so we will compute the time for the plane to reach a velocity of 70 m/s and the position of the plane at this time, giving us the minimum length of the runway.
125 Example 2.12 Calculating the minimum length of a runway SOLVE First we solve for the time required for the plane to reach takeoff speed. We can use the first equation in Synthesis 2.1 to compute this time: We keep an extra significant figure here because we will use this result in the next step of the calculation.
126 Example 2.12 Calculating the minimum length of a runway SOLVE Given the time that the plane takes to reach takeoff speed, we can compute the position of the plane when it reaches this speed using the second equation in Synthesis 2.1: Our final answers are thus that the plane will take 27 s to reach takeoff speed, with a minimum runway length of 940 m.
127 Example 2.12 Calculating the minimum length of a runway ASSESS Think about the last time you flew; 27 s seems like a reasonable time for a plane to accelerate on takeoff. Actual runway lengths at major airports are 3000 m or more, a few times greater than the minimum length, because they have to allow for emergency stops during an aborted takeoff. (If we had calculated a distance far greater than 3000 m, we would know we had done something wrong!)
128 Free Fall When an object moves under the influence of gravity ONLY All objects experiencing free fall (regardless of mass) have the same acceleration (ignoring air all positions
129 Free Fall Acceleration The acceleration of the object in free fall is always negative It is the acceleration due to gravity, afree fall = g = 9.8 m/s 2 vertically downward, i.e., afree fall = m/s 2 = - g
130 QuickCheck Question 2.27 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow s acceleration the greatest? The least? Ignore air resistance; the only force acting is gravity.
131 QuickCheck Question 2.27 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow s acceleration the greatest? The least? Ignore air resistance; the only force acting is gravity. Remember: All objects experiencing free fall (regardless of mass) have the same acceleration (ignoring air all positions
132 QuickCheck Question 2.27 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow s acceleration the greatest? The least? Ignore air resistance; the only force acting is gravity. Answer: Same at all points
133 QuickCheck Question 2.28 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.
134 QuickCheck Question 2.28 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Remember: All objects experiencing free fall (regardless of mass) have the same acceleration (ignoring air all positions = -g = -9.8 m/s 2
135 QuickCheck Question 2.28 An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Remember: All objects experiencing free fall (regardless of mass) have the same acceleration (ignoring air all positions = -g = -9.8 m/s 2
136 The End of Chapter 2 QUESTIONS?
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