June 4, Week 1
Motion Graphs 4th June 2014
Physics 151, Dr. Mark Morgan-Tracy
Today: Chapter 2, Motion Graphs
Please Register your Clicker.
Homework Assignment #1 - Available on class webpage, Due thisFriday, June 6.
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length = meter (m)
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length = meter (m)
• Unit of mass
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length = meter (m)
• Unit of mass = kilogram (kg)
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length = meter (m)
• Unit of mass = kilogram (kg)
• Unit of time
S. I. Units
Motion Graphs 4th June 2014
To compare physical quantities, everyone must use the samesystem of units.
• Physics uses the S. I. system (Systeme International D’unites).
• There are three fundamental units/measurements in S. I.
• Unit of length = meter (m)
• Unit of mass = kilogram (kg)
• Unit of time = second (s)
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
• centi (c) = 0.01 = 10−2
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
• centi (c) = 0.01 = 10−2
• mili (m) = 0.001 = 10−3
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
• centi (c) = 0.01 = 10−2
• mili (m) = 0.001 = 10−3
• micro (µ) = 10−6
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
• centi (c) = 0.01 = 10−2
• mili (m) = 0.001 = 10−3
• micro (µ) = 10−6
• nano (n) = 10−9
S. I. Prefixes
Motion Graphs 4th June 2014
To make more convenient units, the S. I. system has a uniformsystem of prefixes that act as multipliers of powers of ten.
• kilo (k) = 1000 = 103
• mega (M ) = 106
• giga (G) = 109
• tera (T ) = 1012
• centi (c) = 0.01 = 10−2
• mili (m) = 0.001 = 10−3
• micro (µ) = 10−6
• nano (n) = 10−9
• pico (p) = 10−12
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
(b) 3m, 500 cm, 6000µm
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
(b) 3m, 500 cm, 6000µm
(c) 6000µm, 3m, 500 cm
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
(b) 3m, 500 cm, 6000µm
(c) 6000µm, 3m, 500 cm
(d) 6000µm, 500 cm, 3m
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
(b) 3m, 500 cm, 6000µm
(c) 6000µm, 3m, 500 cm
(d) 6000µm, 500 cm, 3m
(e) 3m, 6000µm, 500 cm
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(a) 500 cm, 3m, 6000µm
(b) 3m, 500 cm, 6000µm
(c) 6000µm, 3m, 500 cm
(d) 6000µm, 500 cm, 3m
(e) 3m, 6000µm, 500 cm
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(c) 6000µm, 3m, 500 cm
6000µm = 6000 (10−6) m = 6× 103 (10−6) m = 6× 10−3m = 0.006m
S.I. Exercise
Motion Graphs 4th June 2014
Which of the following correctly lists these distances from smallestto largest?
(c) 6000µm, 3m, 500 cm
6000µm = 6000 (10−6) m = 6× 103 (10−6) m = 6× 10−3m = 0.006m
500 cm = 500 (10−2) m = 5× 102 (10−2) m = 5m
Significant Figures
Motion Graphs 4th June 2014
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
• Usually just the number of digits you see in the number.
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
• Usually just the number of digits you see in the number.
• Exceptions:
• Strings of zeros at the end of large numbers or at thebeginning of small numbers are not significant.
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
• Usually just the number of digits you see in the number.
• Exceptions:
• Strings of zeros at the end of large numbers or at thebeginning of small numbers are not significant.
• Zeroes at the end of all numbers are significant.
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
• Usually just the number of digits you see in the number.
• Exceptions:
• Strings of zeros at the end of large numbers or at thebeginning of small numbers are not significant.
• Zeroes at the end of all numbers are significant.
• When multiplying or dividing, we round to the fewest number ofsignificant figures.
Significant Figures
Motion Graphs 4th June 2014
• Significant Figures = express the accuracy of a measurement.
• Usually just the number of digits you see in the number.
• Exceptions:
• Strings of zeros at the end of large numbers or at thebeginning of small numbers are not significant.
• Zeroes at the end of all numbers are significant.
• When multiplying or dividing, we round to the fewest number ofsignificant figures.
• When adding or subtracting, we round to the fewest places pastthe decimal point.
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min (b)
−13 km
7min= −1.857 km/min
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min (b)
−13 km
7min= −1.857 km/min
(c)−70 km
7min= −10 km/min
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min (b)
−13 km
7min= −1.857 km/min
(c)−70 km
7min= −10 km/min (d)
−57 km
7min= −8 km/min
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min (b)
−13 km
7min= −1.857 km/min
(c)−70 km
7min= −10 km/min (d)
−57 km
7min= −8 km/min
(e)13 km
7min= 2 km/min
Significant Figures Exercise
Motion Graphs 4th June 2014
A car travels from x = −70 km to x = −57 km in 7minutes. What isthe car’s average velocity, in kilometers per minute, recorded to theproper number of significant figures? Use the standard conventionfor direction.
(a)13 km
7min= 1.857 km/min (b)
−13 km
7min= −1.857 km/min
(c)−70 km
7min= −10 km/min (d)
−57 km
7min= −8 km/min
(e)13 km
7min= 2 km/min
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
(b)5 cm
s×
2.54 cm
1 in×
3600 s
1h= 45700 in/hr
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
(b)5 cm
s×
2.54 cm
1 in×
3600 s
1h= 45700 in/hr
(c)5 cm
s×
1 in
2.54 cm×
60 s
1h= 118 in/hr
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
(b)5 cm
s×
2.54 cm
1 in×
3600 s
1h= 45700 in/hr
(c)5 cm
s×
1 in
2.54 cm×
60 s
1h= 118 in/hr
(d)5 cm
s×
1 in
2.54 cm×
1h
3600 s= 5.47× 10−4 in/hr
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
(b)5 cm
s×
2.54 cm
1 in×
3600 s
1h= 45700 in/hr
(c)5 cm
s×
1 in
2.54 cm×
60 s
1h= 118 in/hr
(d)5 cm
s×
1 in
2.54 cm×
1h
3600 s= 5.47× 10−4 in/hr
(e) None of the these
Unit Conversion
Motion Graphs 4th June 2014
We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.
Given that 1.00 in = 2.54 cm, which of the following is the correctconversion of 5 cm/s into in/hr?
(a)5 cm
s×
1 in
2.54 cm×
3600 s
1h= 7090 in/hr
(b)5 cm
s×
2.54 cm
1 in×
3600 s
1h= 45700 in/hr
(c)5 cm
s×
1 in
2.54 cm×
60 s
1h= 118 in/hr
(d)5 cm
s×
1 in
2.54 cm×
1h
3600 s= 5.47× 10−4 in/hr
(e) None of the these
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
Position versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
Position versus time Velocity versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
xPosition versus time Velocity versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time Velocity versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
vx Velocity versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
t
vx Velocity versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
t
vx Velocity versus time
Horizontal Motion
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
t
vx Velocity versus time
Horizontal Motion
Vertical Motion
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
t
vx Velocity versus time
Horizontal Motion
Vertical Motion
t
yPosition versus time
Motion Graphs
Motion Graphs 4th June 2014
In addition to motion diagrams, physicists also like to make graphsto describe motion.
t
xPosition versus time
t
vx Velocity versus time
Horizontal Motion
Vertical Motion
t
yPosition versus time
t
vyVelocity versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
xf
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
xf
tf − ti = ∆t
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
xf
tf − ti = ∆t
xf − xi = ∆x
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
xf
tf − ti = ∆t
xf − xi = ∆x
vx =∆x
∆t
Uniform Motion Position Graph
Motion Graphs 4th June 2014
Walking to right motion diagram: b b b b
Equal spacing between dots because with constant velocity theobject travels the same distance during equal elapsed times.
t
x Position versus time
In uniform motion, theposition graph is a straightline
ti tf
xi
xf
tf − ti = ∆t
xf − xi = ∆x
vx =∆x
∆t
Velocity is the slope of the position versus time graph
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
In math, the slope of linetells you how ”steep” a lineis.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
In math, the slope of linetells you how ”steep” a lineis.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
In Physics, the slope of line is theratio of the change in two physicalquantities.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
t(s)
x(m)
In Physics, the slope of line is theratio of the change in two physicalquantities.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
t(s)
x(m)
In Physics, the slope of line is theratio of the change in two physicalquantities.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
t(s)
x(m)
1 2 3
15
30
In Physics, the slope of line is theratio of the change in two physicalquantities.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
t(s)
x(m)
1 2 3
15
30
∆t = 1 s
∆x = 15m
In Physics, the slope of line is theratio of the change in two physicalquantities.
Math and Physics Slopes
Motion Graphs 4th June 2014
In Physics, slopes have units and don’t necessarily correspond tothe steepness of the line on the drawing.
0 1 2 30
1
2
run
rise
In math, the slope of linetells you how ”steep” a lineis.Slope: m =
rise
run=
1
1= 1
t(s)
x(m)
1 2 3
15
30
∆t = 1 s
∆x = 15m
In Physics, the slope of line is theratio of the change in two physicalquantities.
Slope = Velocity: vx =∆x
∆t=
15m
1 s= 15m/s
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
t
x(b)
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
t
x(b)
t
x(c)
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
t
x(b)
t
x(c)
t
x(d)
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
t
x(b)
t
x(c)
t
x(d)
t
x(e)
Position-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctposition-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
x(a)
t
x(b)
t
x(c)
t
x(d)
t
x(e)
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.
t
x(e)
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.
t
x(e)
Man starts to the right of origin.Walks to left with constantspeed. Passes origin. Stands inplace.
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.
t
x(e)
Man starts to the right of origin.Walks to left with constantspeed. Passes origin. Stands inplace.
Crosses Origin
Position-Graph Followup
Motion Graphs 4th June 2014
t
x(a)
Man walks to the right withconstant speed the whole time.
t
x(c)
Man stands to the right oforigin, magically appears to leftof orgin, stands there.
t
x(d)
Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.
t
x(e)
Man starts to the right of origin.Walks to left with constantspeed. Passes origin. Stands inplace.
Crosses Origin
Stands in Place
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
Velocity versus time
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
vxVelocity versus time
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
t
vxVelocity versus time
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
t
vxVelocity versus time
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
t
vxVelocity versus time
Unchanging vx
Uniform-Motion-Velocity Graph
Motion Graphs 4th June 2014
The simplest graph is the velocity versus time for uniform motion.
Uniform Motion - Constant velocity motion.
Walking to right motion diagram: b b b b
t
vxVelocity versus time
Unchanging vx In uniform motion, velocityis a horizontal line
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
t
vx(b)
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
t
vx(b)
t
vx(c)
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
t
vx(b)
t
vx(c)
t
vx(d)
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
t
vx(b)
t
vx(c)
t
vx(d)
t
vx(e)
Velocity-Graph Exercise
Motion Graphs 4th June 2014
A man walks some distance to the right with constant speed,immediately turns around and walks back to his starting point withthe same speed. Which of the following is the correctvelocity-versus-time graph?
Motion Diagram:b b b b
b b b b Same Point
t
vx(a)
t
vx(b)
t
vx(c)
t
vx(d)
t
vx(e)
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Man walks to the right withconstant speed the whole time
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Man walks to the right withconstant speed the whole time
t
vx(b)
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Man walks to the right withconstant speed the whole time
t
vx(b)
Man speeds up then the manslows down. Going to the rightthe whole time.
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Man walks to the right withconstant speed the whole time
t
vx(b)
Man speeds up then the manslows down. Going to the rightthe whole time.
t
vx(d)
Velocity-Graph Followup
Motion Graphs 4th June 2014
t
vx(a)
Man walks to the right withconstant speed the whole time
t
vx(b)
Man speeds up then the manslows down. Going to the rightthe whole time.
t
vx(d)
Man goes to the right with constant speed. Manimmediately turns around. Man goes to the leftwith faster speed than before.
Velocity-Graph Followup II
Motion Graphs 4th June 2014
t
vx(e)
Velocity-Graph Followup II
Motion Graphs 4th June 2014
t
vx(e)
Man walks to the right but slowing down. Eventually he turns around.Goes to the left with increasing speed and then maintains constantspeed to the left.
Velocity-Graph Followup II
Motion Graphs 4th June 2014
t
vx(e)
Man walks to the right but slowing down. Eventually he turns around.Goes to the left with increasing speed and then maintains constantspeed to the left.
Turns Around
Velocity-Graph Followup II
Motion Graphs 4th June 2014
t
vx(e)
Man walks to the right but slowing down. Eventually he turns around.Goes to the left with increasing speed and then maintains constantspeed to the left.
Turns Around
Constant Speed to Left
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
Works for Uniform Motion:
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
Works for Uniform Motion:
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
Works for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
Works for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
vx
Works for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
vx
∆t
Works for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
vx
∆t
∆xWorks for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
vx
∆t
∆xWorks for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
t
vxAlso works here:
Position from velocity
Motion Graphs 4th June 2014
To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.
t
vx
vx
∆t
∆xWorks for Uniform Motion:
vx =∆x
∆t⇒ ∆x = vx (∆t)
t
vx
∆t
∆xAlso works here:
Instantaneous velocity
Motion Graphs 4th June 2014
When motion is no longer uniform, velocity changes with time.
Instantaneous velocity
Motion Graphs 4th June 2014
When motion is no longer uniform, velocity changes with time.
Average Velocity:
vav =∆x
∆t=
xf − xi
tf − ti
tells use how fast and in what direction an object went on averageduring the elapsed time ∆t.
Instantaneous velocity
Motion Graphs 4th June 2014
When motion is no longer uniform, velocity changes with time.
Average Velocity:
vav =∆x
∆t=
xf − xi
tf − ti
tells use how fast and in what direction an object went on averageduring the elapsed time ∆t.
Instantaneous velocity, vx - How fast and in what direction for oneinstant of time t.
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
t
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
t
Slope = Velocity at time t
Changing Velocity
Motion Graphs 4th June 2014
When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.
t
x Position versus time
t
To find the velocity at onetime t we use the factthat all curves look straightwhen magnified
t
Slope = Velocity at time tNote: To make thisexact we have to makethe magnification infnite.In calculus, this is calledtaking a derivative.
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
t1 t2
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
b
t1 t2
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
b
t1
b
t2
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
b
Slope = vx at t1
t1
b
t2
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
b
Slope = vx at t1
t1
b
Slope = vx at t2
t2
Changing Velocity II
Motion Graphs 4th June 2014
At different points on the curve, the slopes are different (andtherefore so are the velocities).
t
x
b
Slope = vx at t1
t1
b
Slope = vx at t2
t2
Postive slope values range from 0 to ∞
Horizontal lines have 0 slopeVertical Lines have infinite slope