motion graphs 4th june 2014 - physics.unm.edu€¦ · motion graphs 4th june 2014 physics 151, dr....

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


To compare physical quantities, everyone must use the samesystem of units.

S. I. Units



• Physics uses the S. I. system (Systeme International D’unites).

S. I. Units




• There are three fundamental units/measurements in S. I.

S. I. Units





• Unit of length

S. I. Units





• Unit of length = meter (m)

S. I. Units






• Unit of mass

S. I. Units






• Unit of mass = kilogram (kg)

S. I. Units







• Unit of time

S. I. Units







• Unit of time = second (s)

S. I. Prefixes


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



• kilo (k) = 1000 = 103

S. I. Prefixes



• kilo (k) = 1000 = 103

• mega (M ) = 106

S. I. Prefixes



• kilo (k) = 1000 = 103

• mega (M ) = 106

• giga (G) = 109

S. I. Prefixes



• kilo (k) = 1000 = 103

• mega (M ) = 106

• giga (G) = 109

• tera (T ) = 1012

S. I. Prefixes



• kilo (k) = 1000 = 103

• mega (M ) = 106

• giga (G) = 109

• tera (T ) = 1012

• centi (c) = 0.01 = 10−2

S. I. Prefixes



• 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



• 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



• 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



• 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


Which of the following correctly lists these distances from smallestto largest?

S.I. Exercise



(a) 500 cm, 3m, 6000µm

S.I. Exercise



(a) 500 cm, 3m, 6000µm

(b) 3m, 500 cm, 6000µm

S.I. Exercise



(a) 500 cm, 3m, 6000µm

(b) 3m, 500 cm, 6000µm

(c) 6000µm, 3m, 500 cm

S.I. Exercise



(a) 500 cm, 3m, 6000µm

(b) 3m, 500 cm, 6000µm

(c) 6000µm, 3m, 500 cm

(d) 6000µm, 500 cm, 3m

S.I. Exercise



(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



(c) 6000µm, 3m, 500 cm

6000µm = 6000 (10−6) m = 6× 103 (10−6) m = 6× 10−3m = 0.006m

S.I. Exercise



(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


Significant Figures


• Significant Figures = express the accuracy of a measurement.

Significant Figures



• Usually just the number of digits you see in the number.

Significant Figures




• Exceptions:

• Strings of zeros at the end of large numbers or at thebeginning of small numbers are not significant.

Significant Figures




• Exceptions:


• Zeroes at the end of all numbers are significant.

Significant Figures




• Exceptions:



• When multiplying or dividing, we round to the fewest number ofsignificant figures.

Significant Figures




• Exceptions:



• 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


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




(a)13 km

7min= 1.857 km/min (b)

−13 km

7min= −1.857 km/min




(a)13 km


−13 km


(c)−70 km

7min= −10 km/min




(a)13 km


−13 km


(c)−70 km

7min= −10 km/min (d)

−57 km

7min= −8 km/min




(a)13 km


−13 km


(c)−70 km

7min= −10 km/min (d)

−57 km

7min= −8 km/min

(e)13 km

7min= 2 km/min

Unit Conversion


We use the fact that when multiplying or dividing physical quantitiesthat their units also multiply and divide to simplify unit conversion.

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




(a)5 cm

s×

1 in

2.54 cm×

3600 s

1h= 7090 in/hr

Unit Conversion




(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




(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




(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




(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


In addition to motion diagrams, physicists also like to make graphsto describe motion.

Motion Graphs



Position versus time

Motion Graphs



Position versus time Velocity versus time

Motion Graphs



xPosition versus time Velocity versus time

Motion Graphs



t

xPosition versus time Velocity versus time

Motion Graphs



t

xPosition versus time

vx Velocity versus time

Motion Graphs



t


t


Motion Graphs



t


t


Horizontal Motion

Motion Graphs



t


t


Horizontal Motion

Vertical Motion

Motion Graphs



t


t


Horizontal Motion

Vertical Motion

t

yPosition versus time

Motion Graphs



t


t


Horizontal Motion

Vertical Motion

t

yPosition versus time

t

vyVelocity versus time

Uniform Motion Position Graph


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





t


In uniform motion, theposition graph is a straightline





t



ti tf





t



ti tf

xi





t



ti tf

xi

xf





t



ti tf

xi

xf

tf − ti = ∆t





t



ti tf

xi

xf

tf − ti = ∆t

xf − xi = ∆x





t



ti tf

xi

xf

tf − ti = ∆t

xf − xi = ∆x

vx =∆x

∆t





t



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


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.




0 1 2 30

1

2





0 1 2 30

1

2

run

rise





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




0 1 2 30

1

2

run

rise


rise

run=

1

1= 1

In Physics, the slope of line is theratio of the change in two physicalquantities.




0 1 2 30

1

2

run

rise


rise

run=

1

1= 1

t(s)

x(m)





0 1 2 30

1

2

run

rise


rise

run=

1

1= 1

t(s)

x(m)

1 2 3

15

30





0 1 2 30

1

2

run

rise


rise

run=

1

1= 1

t(s)

x(m)

1 2 3

15

30

∆t = 1 s

∆x = 15m





0 1 2 30

1

2

run

rise


rise

run=

1

1= 1

t(s)

x(m)

1 2 3

15

30

∆t = 1 s

∆x = 15m


Slope = Velocity: vx =∆x

∆t=

15m

1 s= 15m/s

Position-Graph Exercise


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





b b b b Same Point

t

x(a)





b b b b Same Point

t

x(a)

t

x(b)





b b b b Same Point

t

x(a)

t

x(b)

t

x(c)





b b b b Same Point

t

x(a)

t

x(b)

t

x(c)

t

x(d)





b b b b Same Point

t

x(a)

t

x(b)

t

x(c)

t

x(d)

t

x(e)

Position-Graph Followup


t

x(a)



t

x(a)

Man walks to the right withconstant speed the whole time.



t

x(a)


t

x(c)



t

x(a)


t

x(c)

Man stands to the right oforigin, magically appears to leftof orgin, stands there.



t

x(a)


t

x(c)


t

x(d)



t

x(a)


t

x(c)


t

x(d)

Man goes to the right withconstant speed. Man turnsaround. Man goes to the left withfaster speed and crosses origin.



t

x(a)


t

x(c)


t

x(d)


t

x(e)



t

x(a)


t

x(c)


t

x(d)


t

x(e)

Man starts to the right of origin.Walks to left with constantspeed. Passes origin. Stands inplace.



t

x(a)


t

x(c)


t

x(d)


t

x(e)


Crosses Origin



t

x(a)


t

x(c)


t

x(d)


t

x(e)


Crosses Origin

Stands in Place

Uniform-Motion-Velocity Graph


The simplest graph is the velocity versus time for uniform motion.




Uniform Motion - Constant velocity motion.






Velocity versus time






vxVelocity versus time






t







t


Unchanging vx






t


Unchanging vx In uniform motion, velocityis a horizontal line

Velocity-Graph Exercise


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?


b b b b Same Point





b b b b Same Point

t

vx(a)





b b b b Same Point

t

vx(a)

t

vx(b)





b b b b Same Point

t

vx(a)

t

vx(b)

t

vx(c)





b b b b Same Point

t

vx(a)

t

vx(b)

t

vx(c)

t

vx(d)





b b b b Same Point

t

vx(a)

t

vx(b)

t

vx(c)

t

vx(d)

t

vx(e)

Velocity-Graph Followup


t

vx(a)



t

vx(a)

Man walks to the right withconstant speed the whole time



t

vx(a)


t

vx(b)



t

vx(a)


t

vx(b)

Man speeds up then the manslows down. Going to the rightthe whole time.



t

vx(a)


t

vx(b)


t

vx(d)



t

vx(a)


t

vx(b)


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


t

vx(e)



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.



t

vx(e)


Turns Around



t

vx(e)


Turns Around

Constant Speed to Left

Position from velocity


To find position from velocity, we use the fact that displacement isthe area under the velocity-versus-time graph.




Works for Uniform Motion:




t

vx





t

vx


vx =∆x

∆t⇒ ∆x = vx (∆t)




t

vx

vx


vx =∆x

∆t⇒ ∆x = vx (∆t)




t

vx

vx

∆t


vx =∆x

∆t⇒ ∆x = vx (∆t)




t

vx

vx

∆t

∆xWorks for Uniform Motion:

vx =∆x

∆t⇒ ∆x = vx (∆t)




t

vx

vx

∆t


vx =∆x

∆t⇒ ∆x = vx (∆t)

t

vxAlso works here:




t

vx

vx

∆t


vx =∆x

∆t⇒ ∆x = vx (∆t)

t

vx

∆t

∆xAlso works here:

Instantaneous velocity


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.




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


When velocity is changing, position versus time is now a curve.Instantaneous velocity is still the slope of the graph.

Changing Velocity



t


Changing Velocity



t


t

Changing Velocity



t


t

To find the velocity at onetime t we use the factthat all curves look straightwhen magnified

Changing Velocity



t


t


t

Changing Velocity



t


t


t

Slope = Velocity at time t

Changing Velocity



t


t


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


At different points on the curve, the slopes are different (andtherefore so are the velocities).




t

x




t

x

t1 t2




t

x

b

t1 t2




t

x

b

t1

b

t2




t

x

b

Slope = vx at t1

t1

b

t2




t

x

b

Slope = vx at t1

t1

b

Slope = vx at t2

t2




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

motion graphs 4th june 2014 - physics.unm.edu€¦ · motion graphs 4th june 2014 physics 151, dr....

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