Page 147 - Calculus Workbook For Dummies

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Chapter 8: Using Differentiation to Solve Practical Problems

A Day at the Races: Position,

Velocity, and Acceleration

The most important thing to know about this type of problem is that velocity is the
derivative of position and acceleration is the derivative of velocity. The following
points about position, velocity, and acceleration with regard to the chariot race in
Figure 8-1 provide some keys to approaching these problems.

50

25

Start

Finish
Figure 8-1:
A 200-
palameter 25
chariot
race.
100

The finish is 100 palameters from the start as the crow flies, so 100 palameters is
the total displacement. (A palameter is a little-known unit of distance used in
ancient Rome equal to the length of Julius Caesar’s palace — roughly 380 feet.)
Say the start is at (0, 0) on a coordinate system and the finish is at (100, 0). It’s
100 from 0 to 100, of course, so 100 is the total displacement.
Distance is different. You can see that the charioteers backtrack 50 palameters in
the middle of the race. Because there are two extra 50-palameter legs, the total
length of the race is 200 palameters — that’s the distance. Distance is always
positive or zero.
Displacement to the left is negative (in other problems, down would be negative).
When, say, Maximus passes Atlas, his position is 75 palameters from the start.
At Aphrodite, he’s back to only 25 palameters from the start as the crow flies.
Displacement equals final position minus initial position, so from Atlas to
Aphrodite is a displacement of 25 - 75, or –50 palameters.
Velocity is related to displacement, not distance traveled. Velocity has a special
meaning in calculus and physics so forget the everyday meaning of it. Like dis-
placement, if you’re going left (or down), that’s a negative velocity. And here’s a
critical point: When you switch directions, your velocity is zero. Think of a ball
thrown straight up. At its peak, for an infinitesimal moment, it is motionless, so
its velocity is zero.