Chapter 1(wrap up)

Units, Physical Quantities, and Vectors

From Pre-lecture comments Also, I think a basic review of calculus principals of increasing/decreasing velocity and acceleration would

be helpful. Difference between physics definitions and everyday definitions. As in, Speed and velocity same in

everyday but not in physics. The association between all three x(t), v(t) and a(t). It helps me more if I have a mental image in my

head, but until then I feel lost. I found the use of graphs to represent displacement, velocity, and acceleration to be confusing. I would like more clarity, on how to relate an acceleration graph, with a velocity graph. I really like this section can we just make sure to cover the creation of the formulas for constant

acceleration? I would like to discuss negative acceleration how can we tell the particle is slowing down or speeding up?

I would like to discuss how in the first checkpoint question there is a negative time on the graph; this seems to be throwing me off. Is it just an implied extension of the graphed line or can there actually be a negative time?

9/8/14 Physics 218 2

The scalar product

• The scalar product (also called the “dot product”) of two vectors is

• Figures 1.25 and 1.26 illustrate the scalar

product.

cos .

ABA B

Calculating a scalar product

[Insert figure 1.27 here]

• In terms of components,

• Example 1.10 shows how to calculate a scalar product in two ways.

.

z zx x y yA B A B A BA B

Finding an angle using the scalar product

• Example 1.11 shows how to use components to find the angle between two vectors.

The vector product

• The vector product (“cross product”) of two vectors has magnitude

and the right-hand rule gives its direction. See Figures 1.29 and 1.30.

| | sin

ABA B

Calculating the vector product

• Use ABsin to find the magnitude and the right-hand rule to find the direction.

• Refer to Example 1.12.

Calculus Summary

9/8/2014 Physics 218 8

1212

120 where)()(lim

:follows as defined is f(x) function, a of Derivative The

xxxdxdf

xxxfxf

x

Graphically this corresponds to the slope of the tangent to the curve f(x) at the point x.

Figure 2.3

9/8/2014 Physics 218 10

1

12

12

12

120

find we)( of polynomial general afor

lim

find weand process limiting out thiscarry can we)(likefunction polynomialsimple aFor

n

n

x

knxdxdf

kxxf

kxxxxk

xxkxkx

dxdf

kxxf

The integral of a function“the definite integral”

9/8/2014 Physics 218 11

b

a

dxxf

baxf

)(

symbol. following eit with th note will weand , to from )(function theof integral thecall will

what weis This curve.arbitrary an under area theinginvestigatfunctions polynomial on theseoperation second a definecan We

Figure 2.28

9/8/2014 Physics 218 13

integer positiveany n for )(

,)( form, theof polynomial general afor Then

)())(()()(

easily.out workedbecan line"straight " a ,)(like curve polynomial simple aunder area The

11

111

1

2221

b

a21

nn

nn

b

a

nb

a

n

kbkadxkxdxxf

kxxf

abkabkakbabkadxxf

bkxxf

The indefinite integral

9/8/2014 Physics 218 14

nintegratio ofconstant )( 11

1 n

nn kxdxkxdxxf

The connection between differentiation and integration

9/8/2014 Physics 218 15

cxfdxdx

xdfxfdxxfdxd

)()( theand )()(

function. that you toreturn function same the toseriesin appliedwhen function aofation differentiandn integratio of operations The

Calculus summary

9/8/2014 Physics 218 16

cxfdxdx

xdf

ckxdxxf

kxxf

knxdx

xdfkxxf

nn

n

n

n

)()(and

)(

)( polynomial a of integral The

)()( polynomial a of derivative The

11

1

1

Chapter 2

Motion along a straight line

9/8/2014 17Physics 218

Goals for Chapter 2

How to describe straight line motion in terms of Average velocity/Instantaneous velocity Average acceleration/Instantaneous acceleration

How to interpret graphs of position vs time; velocity vs time and acceleration vs time for straight line motion.

How to solve problems for straight line motion with constant acceleration.

How to analyze motion in a straigth line when the acceleration is NOT constant.

9/8/2014 18Physics 218

Displacement Vector

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xxx 12ntdisplaceme

Figure 2.1

Average Velocity

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12

12

tx

ttxxvaverage

Figure 2.2

12

12

tx

ttxxvaverage

Figure 2.3

Instantaneous Velocity

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graph timeersusposition v theofcurve theo tangent t theof slope the toscorrespond this:Note

lim 12

120 dt

dxttxxv tx

Figure 2.7

Figure 2.8

Motion Diagramx-t graph

Average and Instantaneous Acceleration

9/8/2014 Physics 218 27

2

2

12

120

12

12

lim

0 elimit wher in the and

dtxd

dtdx

dtd

dtdv

ttvva

tttvva

xxxtx

xxxave

Figure 2.13a

Figure 2.14a

Remember:The second derivative is related to the “curvature “of the function.