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Chapter -1Speed, Velocity, Change

basic idea #1

calculus is the mathematics of change, and change is mysterious. some things grow imperceptibly... others zoom... hair grows slowly and is suddenly cut... tempera tures rise and fall... smoke curls through the air... planets wheel through space... and time, time never stops...

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think hard about change, and you may reach some pretty strange conclusions. in ancient greece, for example, zeno of elea thought about change and convinced himself that motion is impossible. he reasoned like so:

motion is a change of position over time.

at any instant, no change of position takes place.

therefore, there can be no motion at any instant.

therefore, motion never takes place!

but time is a succession of instants.

hey! how did I get

over here?

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in the late 1600s, roughly 2,000 years after zeno, two other guys had a different idea.

isaac newton and gottfried leibniz looked at the problem this way: even though a moving cannonball goes nowhere in an instant, still it has something that indicates motion.

what it has is velocity, a number. you might say that every object carries around an invisible meter that reads out the object’s speed and direction at all times.

even time moves... it’s so weird... actually, I had the

idea and you stole it!you took the words right out of my mouth...

oh, now I’m beginning to see...

in other words, we can imagine that everything has a sort of speedometer, just like the one in a car (except that this speed-ometer indicates direction too).

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a pretty sharp idea for newton and leibniz to have had, considering that speedometers wouldn’t be invented for another 200 years yet...

how did our two geniuses get the idea? to answer this, let’s explore a car’s speedometer reading.

actually, we want a velocimeter, not a speedometer. a velocimeter looks just like a speedometer, except that it attaches a minus sign to the speed when the car is backing up. velocity is the negative of the speed when you go in reverse.

what’s a speedometer?

forsooth!0

–20

–40

–60

20

40

60

what’s a car?

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to appreciate the difference between speed and velocity, imagine a car moving forward for one hour at a steady rate of 50 km/hr, then turning around and coming back (in a “negative direction”) for another hour at the same speed.

the speed is always 50 km/hr, and the car travels a total distance of 100 km: 50 km going out and 50 km coming back. the distance is the speed times the elapsed time:

total distance = speed≈ elapsed time

the average speed is the total distance divided by the time.

speedav = total distance

            Ø elapsed time

but in terms of velocity, the car moves at 50 KM/hr the first hour, and at –50 km/hr the second hour. THE total change of position is zero—the car ends where it started!

its average velocity is the change of position divided by the elapsed time.

vav = change of position

       Ø elapsed time

in this case,

vav = 0 km

= 0 km/hr       Ø2 hr

quite a difference!

= 100 km

= 50 km/hr            Ø 2 hr= (50 km/hr)≈(2 hr)

= 100 km

say, where’d you learn to drive?

copied you.

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in symbols: if t1 and t2 are any two times, and an object is at position s1 at time t1 and at position s2 at time t2, then the object’s average velocity over the time interval between t1 and t2 is

what does it mean when delta’s velocimeter reads 100 km/hr? for one thing, it must mean that if she were to hold her velocity perfectly steady, then she would go 100 km in one hour, right? (delta has mounted a clock on the roof for clarity.)

vav = s2 – s1

       Øt2 – t1

if I start here at noon...

and we’d go 200 km in 2 hours, 50 km in half an hour, 100t kilo meters in  t hours... a formula that should work even for short time inter vals. at a perfectly steady 100 km/hr, delta goes 1 km in ˙ hour (36 sec-onds), 0.1 km in 0.001 hour (3.6 seconds), and 0.001 km, one meter, in 0.00001 hr, or 0.036 seconds.

er... logical, I guess...

or

s2 – s1 = vav(t2 – t1)

now we need a better driver—someone with a steadier foot—so let’s put my friend delta wye behind the wheel...

yo!

I arrive here at one o’clock!

  t2–t1  s2–s1  (hours) (kilometers)

10  1000  9  900  5  500  1  100  0.5  50  0.1  10  0.01  1  0.001  0.1  0.0001  0.01  0.0000001  0.00001

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that’s if the velocity remains perfectly steady... but in the real world, velocity changes as a car slows down and speeds up. what does the reading mean then? (now she’s added a velocimeter up top as well.)

the answer is a little subtle: you’ve surely noticed that over a very short time period, a speedometer doesn’t change much. even if you floor it, v is nearly constant over a time span of, say, 1/500 sec. a photo taken with a short exposure would show a velocimeter image with virtually no blur.

this was newton’s and leibniz’s

Basic Idea: calculate the ratio (s2–s1)/(t2–t1) over a very short time interval. for all intents and pur poses, this ratio is the velocity at time t1  (and also at t2 , they’re so close!).

what’s a photo?

speeding up

slowing down

VElocity = 0 velocity high

velocity lower

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but newton and leibniz wanted more than an approximation: they wanted the velocity’s exact value... and what’s more, they showed how to get it! forget measurement: they used math, a new kind of math they invented especially for the purpose.

to put it another way, a body’s instantaneous velocity is closely approxi­mated by (s2–s1)/(t2–t1) when t2–t1 is small. (you might wonder how newton and leibniz thought they might actually measure a change of position over a time interval of, say, 0.00001 sec., but never mind that!)

arrhefff! it’s the principle of the thing...

and we’ll call it fluxions!

no. we won’t.

we call it calculus.

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(leibniz’s approach seemed especially fishy: he was happy to divide one thing by another not only when the quantities were small, but also when they were “infinitely small” but

not zero, whatever that meant.)

if a body’s position depends on time according to some formula, then calculus pops out a new, exAct formula for the velocity at any time.

this seemed so magical that more than a few people found it suspicious... weird... based on strange, unfounded assumptions... somehow... wrong...

freak!

you’re almost dividing by zero!

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fishy foundations or not, calculus worked, and it worked beautifully. it was amazingly effective. it produced results!

MANY, ÂŘ¥, many results...

so people put calculus to work... not only finding velocities, but also the rate of change of all kinds of fluctuating quantities. calculus is used everywhere!

eventually, they even fixed the foundations, more or less... unfortunately, we lack the space to explain fully how this was done, or to describe the trouble-some issues raised by calculus... let’s just say that some of zeno’s subtleties remain a challenge to this day...

whoa!

astronomy, communications,

electricity, biology, chemistry, mechanics, statistics, computer science, psychology,

economics...

population dynamics...

hey, man, you worry too much!

yeah, c’mon! whatever works...