Physics 218, Lecture II 1

Dr. David Toback

Physics 218Lecture 2

Physics 218, Lecture II 2

In Class Quiz

Write down the most important “student case study” from the Frequently Asked Questions handout

Physics 218, Lecture II 3

Announcements: WebCT• Having trouble getting started? Try:

– ITS Help sessions – Open access lab/student computing – Instructions on

faculty.physics.tamu.edu/toback/WebCT– email to webct@physics.tamu.edu

• Check your neo email account for announcements • Still working on Math Quiz figures… sorry about

that..• Finish your “Preliminary Course Materials”

Physics 218, Lecture II 4

Due dates coming up•Week 1 (This week):

– Lecture: Chapter 1 (Reading, but nothing due)– Recitation & Lab: Lab 1 (A&B) – Homework due: None

•Week 2 (Next week):– Homework (Monday): Math quizzes– Lecture: Chapter 2– Recitation & Lab: Chapter 1 and Lab 2

•Week 3 (The week after that):– Homework due (Monday): Chapter 1– Lecture: Chapter 3 & 4 – Recitation: Chapter 2 and Lab 3

•Etc..

Physics 218, Lecture II 5

Physics 218, Lecture II 6

Chapter 1: Calculus•Won’t cover the chapter in detail•This is a chapter that is best learned by DOING•We’ll cover it quickly– Lots more examples in Chapter 2– Lots of practice in Math Quizzes on WebCT (when they’re fixed)

Physics 218, Lecture II 7

Where are we going?We want Equations that describe• Where am I as a function of time?

• How fast am I moving as a function of time?

• What direction am I moving as a function of time?

• Is my speed changing? Etc.

Physics 218, Lecture II 8

Physics 218, Lecture II 9

Motion in One Dimension• Where is the car?

– X=0 feet at t0=0 sec– X=22 feet at t1=1 sec– X=44 feet at t2=2 sec

• Since the car’s position is changing (i.e., moving) we say this car has “speed” or “velocity”

• Plot position vs. time– How do we get the

speed from the graph?

Physics 218, Lecture II 10

SpeedQuestions:• How fast is my position changing?

• What would my speedometer read?

Physics 218, Lecture II 11

How do we Calculate the speed?• Define speed: “Change in position during a certain amount of time”

• Math: Calculate from the Slope: The “Change in position as a function of time”– Change in Vertical divided by the Change in Horizontal

– Speed = ΔX/ΔtChange: Δ

Physics 218, Lecture II 12

Constant SpeedEquation of Motion for this example is a straight lineWrite this as:

X = bt• Slope is constant• Velocity is constant

– Easy to calculate– Same everywhere

Position time

Physics 218, Lecture II 13

Moving CarA harder example:

X = ct2

•What’s the speed at t=1 sec?

Want to calculate the “Slope” here

What would the speedometer say?

Physics 218, Lecture II 14

Derivatives• To find the slope at time t, just take the “derivative”

• For X=ct2 , Slope = V =dx/dt =2ct• “Gerbil” derivative method–If X= atn →V=dx/dt=natn-1

– “Derivative of X with respect to t”• More examples

– X= qt2 →V=dx/dt=2qt– X= ht3 →V=dx/dt=3ht2

Physics 218, Lecture II 15

Common MistakesThe trick is to remember what you are taking the derivative “with respect to”

More Examples (with a=constant):• What if X= 2a3tn?

– Why not dx/dt = 3(2a2tn)?– Why not dx/dt = 3n(2a2tn-1)?

• What if X= 2a3?– What is dx/dt?– There are no t’s!!! dx/dt = 0!!!– If X=22 feet, what is the velocity? =0!!!

Physics 218, Lecture II 16

Going the other way: Integrals

• What if you know how fast you’ve been going and how long you’ve been driving

• How can you figure out how far you’ve gone?

• What would your car’s odometer read?

Physics 218, Lecture II 17

Getting the Displacement from Velocity

• If you are given the speed vs. time graph you can find the total distance traveled from the area under the curve:

X=V0t + ½at2

• Can also find this from integrating…

∫=t

ovdtx

Slope is constant =Constant acceleration

Physics 218, Lecture II 18

Definite and Indefinite Integrals

cacbct|cdt

constants) are c anda, b, (assuming s:end and begins nintegratio of region my where know I If

cdt

b)d(ct equation the of side right the to added

is and constant arbitrary an is b where b ct dt )c(

constc For .itiveanti-deriv an is integral an ways many Inintegral? an of Value the calculate you to How

btat

b

a−==

=+

⇒

+=⇒

=

==∫

∫

Physics 218, Lecture II 19

Some Integrals

( ) ( )

m1m

t

1mt

dtdm

dtd

at0)(a )1(m

c )(a dtat"derivitive-nti"a Check

11)(m

1m

=

+•+=

+=

+

+

−+

+

∫

∫∫

+=

+=

+

+ c )a( dt at

c at a dt :general more this Make

1mtm 1m

Physics 218, Lecture II 20

Our Example

221

0

t0

221

0

t

o 0

t

o0

attv )|att(v

dtat)(v

vdtx-x

+=

+=

+=

=

∫∫

Physics 218, Lecture II 21

For Next Week•Before Lecture:– Read Chapter 2– Math Quizzes due Monday

•In Lecture– Cover Chapter 2

•Recitation, Lab and Homework:– Start Chapter 1 problems and exercises before recitation

– Read your lab materials beforelab