Physics 140: University of

Michigan

September 9, 2013

Prof. Tim McKay

Racquetball Striking a Wall Copyright: Loren M. Winters

Mt. EtnaAndrew Davidhazy

Scalars and vectors

• Scalar: just a size, a magnitude – Price– Mass– Age

• Combining scalars: simple addition of magnitudes

• Vector: both a magnitude and a direction– Force– Displacement (a trip)– Wind

• Combining vectors, both magnitude and direction must be considered

Vectors

• Things with both magnitudes and directions

• For equality both magnitude and direction must be equal

• Multiplying by scalars changes only magnitude

• Displacement vectors are instructions: go this far in this direction.

• They are not attached to particular points (go from A to B)

• Addition of vectors just means do one, then the next…

Vector addition: geometric

To combine two vectors, do one, followed by the other…

+ = Resultant Vector

To subtract, just add the opposite…

- =+ =

Resultant Vector

a+b=b+a=c“commutativity”

Odd feature of vectors

Adding up a set of non-zero vectors can lead to a sum which is zero… Examples: • paddling upstream• canceling forces

+

Demonstration

Given the following three vectors:

AB

C

Which of the following could be the sum A+B+C?

1:

2:

3:

4:

Finding the ‘components’ of vectors

Any vector can be written as a sum of other vectors. It is often useful to write a vector as a sum of ‘component’ vectors, each of which is parallel to a chosen coordinate axis.

x

yv

vx

Vy

This representation is powerful because it allows particularly easy addition. You can add two (or more) vectors simply by adding the magnitudes of their x and y (and z if necessary) components.

For this case we can write: |vx| = |v|cos() (x component) |vy| = |v|sin() (y component)And |v| = (|vx|2 + |vy|2) (magnitude) = tan-1(vy / vx) (direction)

To add vectors by components, “resolve” each vector into components, then add the components to produce the resultant vector….

v1 = 3xunit + 5yunit

v2 = 2xunit – 3yunit

v1 + v2 = 5xunit + 2yunit

Adding and subtracting by components is just much

easier…

Kinematics: a description of motion

• “The science of pure motion, considered without reference to the matter or objects moved, or to the force producing or changing the motion.” In French: “cinematique”…

• Start simply– Point object– One dimensional

motion, back and forth along a line…

• Note position “si” at every instant “ti”: this describes the motion

0 2 4-2-4

Positions at various times

• Positions si are not distances, but locations at instants ti

• Displacements

s12 = s2 – s1

• Intervals

t12 = t2 – t1

• Often represent this in a ‘position-time’ graph

s

t

Records position s at every instant t, and provides a visualization of the motion

What do we need to describe?

• The whole story is there in the path s(t): complete description

• To explain this motion, we need to think about both how fast it moves, and especially about how that motion changes

Moving slow

Moving fast

Speeding up

Slowing down

Moving slow

s

t

A physics student starts (t=0) at the library in the illustration, lingers there for a moment, and then walks along to the gym, and stays there for a moment. She then runs quickly to the Arb, and then slowly strolls back to the library. Which of the following position-time graphs represents her motion?

0

Library GymArb

1

2

3

4

+-

Rate of motion: speed

• How fast? How much distance in how much time?

v = s / t• Lingo:

– “average speed” over finite time interval

– “instantaneous speed” in the limit t=>0 (calculus invented to describe this…)

• Units: distance/time => meters / second

• Useful fact: 1 m/s = 2.24

mph45 m/s = 100

mph

Some speeds

• Sea floor spreading: 1x10-9 m/s (~3cm/yr)• Grass growing: 5x10-8 m/s• Glacier: 3x10-6 m/s• Walking: 1.3 m/s• Car: 25 m/s• Sound: 330 m/s• Earth's motion around the sun: 2.9x104 m/s• Sun's motion around the Milky Way center: 2.2x105 m/s• Speed of an electron orbiting in Hydrogen: 2x106 m/s• Speed of light: 2.998x108 m/s (~1ft/ns)

Position-time and speed-time graphs

• Speed:

• Speed is the slope of position-time graph

• Speed at each instant can be displayed in a speed-time graph

s

t

v

t

Speed determined from the slope of the position-time graph

dt

ds

t

stv t

0lim

Position from the speed-time graph

• Since v=s/t, distance in each interval is s=vt

• Total distance traveled is just the sum of all intervals

• This is the area under the speed-time curve: the integral

f

i

t

t

dttvs

v

t

s

t

Note: the speed-time graph constrains only the interval, how far it has traveled. The starting point is arbitrary.

A computational example:

tdt

dstv

ttts

2.00.5

1.00.50.15 2

What would be the distance traveled from t = 0 to t = 4s?

4

0

4

02

4

0

04

04

6.216.1201.00.52.00.5

6.21)15()6.12015()0()4(

tttdttvs

sss

Note that position at t = 0 cancels out…

Changing motion?

• To describe changing speed: “acceleration”

• Defined like speed

• Slope of the speed-time graph

• Acceleration at each instant can also be displayed

dt

dv

t

vta t

0lim

s

t

v

t

a

t

Speeding up

Slowing down

Variations…

• Consider the coordinate system:

• Various motions are possible. Speed and acceleration can be in the same direction or not…

0 2 4-2-4

Description Sign of speedSign of

acceleration

Moving right, speeding up

+ +

Moving right, slowing down

+ -

Moving left, speeding up

- -

Moving left, slowing down

- +

This graph shows the position-time graph for two trains running on parallel tracks. Which is true?

s

t

1. At time tb, both trains have the same speed2. Both trains accelerate all the time3. Both trains have the same speed at some time

before tb

4. Somewhere on the graph, both have the same acceleration

tb

Relationships

• Differential: • Integral

2

2 )()()(

)()(

)(

dt

tsd

dt

tdvta

dt

tdstv

ts

)(

)(

)(

ta

dttavtv

dttvsts

f

i

f

i

t

t

i

t

t

i

Slopes…. Areas under curves….

Which of the four example speed-time curves corresponds to the acceleration time curve shown?

a

t

v

t

v

t

v

t

v

t

1

2

3

4

Particular motions

• Constant position (pretty boring)

• Constant speed: continuous motion (slightly less boring)

i

ii

vtv

tvsts

)(

)(

s

t

v

t

a

t

Particular motions

• Continually changing motion: constant acceleration

• What happens: object moves faster and faster, leaves the area ‘quickly’…

• Usually see this only for brief periods…

s

t

v

t

a

t

How practical are these?

• Constant position is not worth talking about…no change, no physics

• Constant velocity is fairly common: we will see that it doesn’t require explanation!

• Constant acceleration– Not very common– Simple, and

occasionally happens, for a short while

– Interesting! This requires an explanation…

Examples of constant a

• Accelerating car, train, etc…

• Free fall: unimpeded motion of a dropped object

• How do objects fall?• Aristotle: gravity and

levity…heavy objects fall faster…authority

• Galileo reads the book of nature, and uses:– Idealization: remove

air resistance– Experiment: ‘dilute’

acceleration with inclined plane

– Thought experiments: balls of clay

Demo

Paper demo

???

What Galileo found

• Gravity accelerates all objects at the same rate independent of their properties!

• An elephant and a feather fall in the same way (if you remove distractions like air friction)

• This ‘acceleration due to gravity’ has a value of about 9.8 m/s2 near the Earth’s surface

• It always happens towards the center of the Earth

Demo