ap physics c - calculus

7/25/2019 AP Physics C - Calculus

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The Basics of Physics withCalculus

AP Physics C


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Pythagoras started it all6th

CenturyPythagoras first got interested in music when

he was walking past a forge and heard thatthe sounds of the blacksmiths' hammers

sounded good together. He talked to theblacksmiths and found out that this wasbecause the anvils they were using werescaled down copies of each other: one fullsize, one half size, and one two thirds size.He got an idea that maybe a simple fractionrelationship in the size of two instruments iswhat makes them sound good together.

This idea led to the belief that mathematicswas tied to the physical world.


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Galileo GalileiGalileo too grew up around music as

his father was a musician. His father,

however, felt that the PythagoreanHarmonies were too simple for thenow emerging Italian Renaissance.

Galileo, like his father, was dealingwith this same idea as well. Hisstudies with kinematics (motion)

were too abstract and too simple toexplain complicated motion.


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Differential Calculus More sophisticated!

25 years later Isaac Newton and Gottfried Leibnizdeveloped a sophisticated language of numbers andsymbols called Calculus based on work. Newton

began his work first but it was Leibniz who firstpublished his findings. Both led the other towardsaccusations of plagiarism.


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What is calculus?Calculus is simply very advanced algebra andgeometry that has been tweaked to solvemore sophisticatedproblems.

Question: How much energy does the man use to push the crate up the

incline?


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The regular wayFor the straight incline, the man

pushes with an unchanging

force, and the crate goes upthe incline at an unchangingspeed. With some simplephysics formulas and regularmath (including algebra andtrig), you can compute howmany calories of energy are

required to push the crate upthe incline.


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The calculus wayFor the curving incline, on the otherhand, things are constantly changing.The steepness of the incline ischanging and not just in

increments like its one steepness forthe first 10 feet then a differentsteepness for the next 10 feet itsconstantlychanging. And the manpushes with a constantly changing

force the steeper the incline, theharder the push. As a result, theamount of energy expended is alsochanging, not every second or everythousandth of a second, butconstantly changingfrom onemoment to the next. Thats whatmakes it a calculus problem.


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What is calculus?It is a mathematical way to express something that isCHANGING! It could be anything??

But here is the cool part:

Calculus allows you to ZOOM in on a small part of the problem and apply

the regular math tools.


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Regular math vs. Calculus


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Learn the lingo!Calculus is about rates of change.

A RATEis anything divided by time.

CHANGEis expressed by using the Greekletter, Delta, .

For example: Average SPEED is simply the

RATE at which DISTANCE changes.


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The Derivativeaka.The SLOPE!Since we are dealing with quantities that are changing it may be

useful to define WHAT that change actually represents.

Suppose an eccentric pet ant is constrained to move in onedimension. The graph of his displacement as a function oftime is shown below.

t

x(t)

t + t

x(t +t)

A

B

At time t, the ant is located at Point

A. While there, its position coordinateis x(t).

At time (t+ t), the ant is located atPoint B. While there, its position

coordinate isx(t + t)


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The secant line and the slope

t

x(t)

t + t

x(t +t)

A

B

Suppose a secant line is drawn between points A and B. Note: The slope of

the secant line is equal to the rise over the run.


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The Tangent lineREAD THIS CAREFULLY!

If we hold POINT A fixed while allowing t tobecome very small. Point B approaches Point A

and the secant approaches the TANGENT to thecurve at POINT A.

t

x(t)

t + t

x(t +t)

A

B

We are basically ZOOMING in at point A where upon inspection the line

APPEARS straight. Thus the secant line becomes a TANGENT LINE.

A

A


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The derivativeMathematically, we just found the slope!

linetangentofslope)()(

lim

linesecantofslope)()(

0

12

12

=

+

=

+=

=

t

txttxt

txttx

xx

yyslope

t

Lim stand for "LIMIT" and it shows the delta t approaches zero.

As this happens the top numerator approaches a finite #.

This is what a derivative is. A derivative yields a NEW function that defines

the rate of change of the original function with respect to one of itsvariables. In the above example we see, the rate of change of "x" withrespect to time.


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The derivative

In most Physics books, the derivative is writtenlike this:

Mathematicians treat dx/dt as a SINGLE SYMBOL which means find thederivative. It is simply a mathematical operation.

The bottom line: The derivative is

the slope of the line tangent to apoint on a curve.


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Derivative exampleConsider the function x(t) = 3t +2What is the time rate of change of the function? That is, what is the NEW

FUNCTION that defines how x(t) changes as t changes.

This is actually very easy! The entire equation is linear and looks like y =mx + b . Thus we know from the beginning that the slope (thederivative) of this is equal to 3.

Nevertheless: We will follow through by using the definition of thederivative

We didn't even need to

INVOKE the limit becausethe delta t's cancel out.

Regardless, we see thatwe get a constant.


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ExampleConsider the function x(t) = kt3, where k = proportionality

constant equal to one in this case..

What happened to all the

delta t's ? They went to

ZERO when we invokedthe limit!

What does this all mean?


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The MEANING?

For example, if t = 2 seconds, using x(t) = kt3

=(1)(2)3

=8 meters.

The derivative, however, tell us how our

DISPLACEMENT (x) changes as a function ofTIME (t). The rate at which Displacement changesis also called VELOCITY. Thus if we use our

derivative we can find out how fast the object is

traveling at t = 2 second. Since dx/dt =3kt2=3(1)(2)2= 12 m/s

23 3)( ktdt

ktd=


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THERE IS A PATTERN HERE!!!!

Now if I had done the previous example withkt2, I would have gotten 2t1

Now if I had done the above example withkt4, I would have gotten 4t3

Now if I had done the above example withkt5, I would have gotten 5t4


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Lets cheat..Forgive us Mrs. Grasso!


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Common calculus derivative rules

ap physics c - calculus

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