Tuesday, September 15, 2009

Calculus in 5 minutes (part 2)

In the first post of this series I mentioned that a car's odometer and speedometer perform related functions :), and these functions are a good way to think about the fundamental operations of calculus. Suppose a car is going in a straight line (say, on the Bonneville salt flats). We need to deal only with the positive x-axis.

Distance (often labeled s) is a function of time:

s = f(t)

If the velocity is constant and we start from s = 0 then:

s = v * t

Distance equals velocity times time. (60 miles per hour for 2 hours = 120 miles).

Now, velocity might be changing, it might be a function of time. Let's say we'll figure out how it changes later, and just write in a general way:

v = f(t)

If the velocity is changing, we call the rate of change in velocity the acceleration. (It is the slope of the curve for v as a function of t). If the acceleration is constant (like gravity), then:

v = a * t

For a car, imagine increasing the pressure on the gas pedal steadily so that after 1 second you are going 10 mph, after 2 seconds 20 mph, after 3 seconds 30 mph. If we continue at the same rate, we'll accelerate from 0 to 60 mph in 6 seconds.

As I said before, there is a special trick symbolized with a prime mark placed next to the s (the physicists put a dot on top). The trick is called differentiation:

s' = ... something.

What it is depends on the exact form of s as a function of time. If the acceleration is constant:

s = 1/2 a t2

Do you remember the rule from last time?

y  = c xn
y' = n c xn-1

Or, using the time and distance symbols:

s  = 1/2 a t2
s' = v = a t
s'' = a

So that is where the factor of 1/2 comes from in the equation involving acceleration. It is needed to cancel the the 2 that comes out of the exponent when we differentiate. One way to think about this is to say that the velocity after a period of constant acceleration is the average of the initial acceleration and the final acceleration, times the time. But this differentiation mumbo jumbo will work even if the velocity and acceleration are more complicated functions of time.

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