physics 218, lecture iii1 physics 218 lecture 3 dr. david toback

Physics 218, Lecture III 1

Physics 218Lecture 3

Dr. David Toback


Checklist for Today•Things that were due last Thursday:

– Chapter 1 reading– Read all handouts from web page

•Things that are due yesterday (Monday):– WebCT warm-ups (FCI, Math Assess, etc…)– Math Quizzes 1 through 10

•Things that are due today:– Reading for Chapter 2– Chapter 2 Lecture Questions

•For this week and/or due next Monday:– Recitation: Lab materials, start Ch. 1 on WebCT– All HW1 problems on WebCT due Monday


Describing Motion

Interested in two key ideas:

• How objects move as a function of time

–Kinematics

–Chapters 2 and 3

• Why objects move the way they do

–Dynamics

–Do this in Chapters 4 and 5


Chapter 2: Motion in 1-Dimension

• Today: Velocity & Acceleration–Equations of Motion

–Some calculus (derivatives)

• Thursday:–More calculus (integrals)

–Problems


Notes before we begin

• This chapter is a good example of a set of material that is best learned by doing examples

• We’ll do some examples today

• Lots more next time…


Equations of Motion

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 its velocity changing? Etc.


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 “velocity” or “speed”

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

velocity from the graph?


Velocity

Questions:

• How fast is my position changing?

• What would my speedometer read?

• What is my instantaneous Velocity?


How do we Calculate Velocity?

• Define Velocity: “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

– Velocity = XtChange:


Constant Velocity

Equation of Motion for this example is a straight line

Write this as:X = bt

• Slope is constant• Velocity is constant

–Easy to calculate–Same everywhere


Moving Car

A harder example:

X = ct2

•What’s the velocity at t=1 sec?

Want to calculate the “Slope” here


Math Digression: 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


Common Mistakes

The 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!!!


Check: Constant Position

– X = C = 22 feet

– V = slope = dx/dt = 0• Check


Check: Constant Velocity

• Car is moving– X=0 feet at t0=0 sec– X=22 feet at t1=1 sec– X=44 feet at t2=2 sec

• What is the equation of motion?

• X = bt with b=22 ft/sec• V = dX/dt

V= b = 22 ft/sec• Check


Check: Non-Constant Velocity

• X = ct2 with c=11 ft/sec2

• V = dX/dt = 2ct

• The velocity is:

• “non-Constant”

• a “function of time”

• “Changes with time”– V=0 ft/s at t0=0 sec

– V=22 ft/s at t1=1 sec

– V=44 ft/s at t2=2 sec


Acceleration

• If your velocity is changing, you are “accelerating”–You hit the accelerator in your car to

speed up at a stop light• (Ok…It’s true you also hit it to stay at constant

velocity, but that’s because friction is slowing you down…we’ll get to that later…)

• How quickly is the velocity changing? That’s our Acceleration


Acceleration

• Acceleration is the “Rate of change of velocity”

• Said differently: “How fast is the Velocity changing?” “What is the change in velocity as a function of time?”

dtdV

ttVV

ΔtΔVAccel

12

12


Example

You have an equation of motion where:

X = X0 + V0t + ½at2

where X0, V0 , and a are constants.

What is the velocity and the acceleration?

V = dx/dt = 0 + V0 + at

• Remember that the derivative of a constant is Zero!!

Accel = dV/dt =d2x/dt2 = 0 + 0 + a


Position, Velocity and Acceleration

• All three are related– Velocity is the derivative of position with

respect to time– Acceleration is the derivative of velocity with

respect to time– Acceleration is the second derivative of

position with respect to time

• Calculus is REALLY important• Derivatives are something we’ll come back

to over and over again


Important Equations of Motion

If the acceleration is constant

Position, velocity and Acceleration are vectors. More on this in Chap 3

221

00

0

tatv x x

tav v


Conceptual Example

• If the velocity of an object is zero, does it mean that the acceleration is zero?

• If the acceleration is zero, does that mean that the velocity is zero?


Car Crash Test Design

You are designing a crash test setup for a car maker. You can accelerate a car from rest with a constant acceleration of 1.00 m/s2 so you can make the car crash into a wall. (This is the last time you will see numbers in a problem in lecture).

1. If the path is 200m long, what is the velocity of the car just before/as it hits the wall?

2. For the same acceleration, if you want the car to hit the wall with a speed of 30m/s (about 60 mi/hr), what minimum length must you have?


Next Time• Textbook Reading and Reading Questions:

– None (Chap 3 assigned on Thursday)• Homework: Math Quizzes and Chap 1

– Math Quizzes were due Monday – Work all Ch 1 probs before recitation– Start WebCT for Ch 1 before recitation– Chapter 1 HW due next Monday

• Recitation: Ask TA for help on hard HW problems

• Lab: Read VP Manual before lab • Thursday: more example problems


Decelerating CarYou are driving a car along a straight highway when you put on the brakes. The initial velocity is 15.0m/s to the right, and it takes 5.0s to slow the car down until it is moving at 5.0m/s to the right.What is the car’s average acceleration?


Examples

• Can a car have uniform speed and non-constant velocity?

• Can an object have a positive average velocity over the last hour, and a negative instantaneous velocity?


Constant Velocity

12

12Velocity Velocity Average ttXX

ΔtΔXv

This example:X = bt

• Slope is constant• Velocity is constant

–Easy to calculate–Same everywhere


More Questions on the Car Crash

• What is the distance traveled?

• What is the total displacement?

• What is the average speed?

• Is the average speed the same as the average velocity?

• What is the instantaneous velocity at all times?


Reference Frames

Frame of reference:

• Need to refer to some place as the origin

• Draw a coordinate axis– We define everything from here

– Always draw a diagram!!!


If the motion started

here, call this x0

Displacement

Where are you? I.e, What is your displacement?

Well…relative to where? Example: I’m 10 blocks north east of Kyle field

What do we need to know? Where does the motion start? x0?

• x0 is relative to the origin• x0 meters from the origin

When does the motion start? t0?


Vectors vs. Scalars

Scalar Distance traveled is 100m

Vector Displacement is 40m East

Let’s say we traveled on a path like in the figure

• Distance traveled from the origin is a Scalar (like your car odometer).

• Displacement from the origin is a Vector– Has a distance and a direction

from the origin

• Speed is a scalar• Velocity is a vector

– Negative distance? – Displacement?


Another reason to care about vectors

• It turns out that nature has decided that the directions don’t really care about each other.

• Example: You have a position in X, Y and Z. If you have a non-zero velocity in only the Y direction, then only your Y position changes. The X and Z directions could care less. (I.e., they don’t change).

Represent these ideas with Vectors


Acceleration

• An object is accelerating if it’s “velocity is changing as a function of time”– Acceleration = dv/dt

• Acceleration and velocity can be pointing in different directions– How?

• What is the difference between average acceleration and instantaneous acceleration?


If the motion started

here, call this x0

Displacement

Where are you? I.e, What is your displacement?

Well…relative to where? Example: I’m 10 blocks north east of Kyle field

What do we need to know? Where does the motion start? x0?

• x0 is relative to the origin• x0 meters from the origin

When does the motion start? t0?


Average Velocity

• Average speed

• Average velocity

Total time = 10secAvg Speed = 100m/10s = 10m/sAvg Velocity = (40m East)/10s

= 4m/s East

Time Total

Traveled Distance

Time Total

tDisplacmen Total

Time Total

Position) (Initial-Position) (Final


Instantaneous Velocity

Average and Instantaneous Velocity

• Average is “over a period of time” – I.e., How many miles you traveled in a day

• Instantaneous is how fast are you going “right now”

• Car example: – Instantaneous is more like your speedometer.

– Average is taking how far you traveled in the last hour and and dividing by an hour (includes the stops at the gas station)


Instantaneous Cont…

• V=x/t (use total change in x, t: average)

• (instantaneous)

• Magnitude of instantaneous velocity is always the same as the instantaneous speed– Why? In the last example, is the average velocity

the same as the average speed?

• Distance and displacement become identical in the limit that they become infinitesimally small

dtdx

tX

0tLimit 0tLimit V


Calculus 1

• Why are we doing math in a Physics class?

• Believe it or not, Calculus and Classical mechanics were developed around the same time, and they essentially enabled each other.

• Calculus basically IS classical mechanics

• Bottom line: If you can’t do Calculus you can’t REALLY do physics. – It’s true you can do some simple problems


Advice

• You really need to be comfortable differentiating!

• If you aren’t, do lots of problems in a introductory calculus book and take lots of math quizzes

• The “rate” at which things “change” will be really big in everything we do

• If you are struggling with the problems in the handout get help now

• This stuff is going to go by quickly!


Overview• I’m not going to teach you calculus• The goals are:

– Teach (hopefully remind) you about how to think about how things “change as a function of time”

– Teach you how to take a derivative (and why you take derivatives) so you can get by until you get to it in your calculus class

• Diagrams are vital again!• Units here will really help (there is a good

example of this in problem 1-9 on the Calculus handout).


Some Notation

• Let’s do some definitions

• Define “define” – Example: t0 0 sec

– We can always make a definition, the idea is to make one that is “useful”

– Another example: X = 22 meters X0

• Define as “the change in”

physics 218, lecture iii1 physics 218 lecture 3 dr. david toback

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