Venn, John 523

of the object at time t, then the quantity f(t+ h) – f(t)

represents the change in position of the object from time

tto a later time t+ h. The average velocity of the object

over this time period is given by:

Computing the average velocity over smaller and

smaller time intervals, that is, in taking smaller and

smaller values of hin the formula above, gives the

“actual” velocity of the object at time tas read by a

speedometer, say. We have:

This, of course, is the formula for the

DERIVATIVE

of the

distance function f(t).

For an object moving in a straight-line path,

the instantaneous velocity of the object is the

derivative of the distance formula. That is,

instantaneous velocity is the instantaneous rate

of change of position.

Now approximate the area under a velocity-vs.-time

graph as a collection of narrow rectangles. Since, as we

have seen, the area of each rectangle represents the dis-

tance traveled over a small period of time, the sum of

these areas gives an approximation of total distance

traveled by the object. Using narrower and narrower

rectangles gives better and better approximations. In

the

LIMIT

we have:

For an object moving in a straight-line path, the

INTEGRAL

of the velocity function gives the total

distance traveled. That is, the distance traveled

is the area under the velocity-vs.-time graph.

If an object in motion follows a curved path, then

one usually assigns to velocity not just a magnitude,

but also a direction of motion. That is, velocity is con-

sidered a

VECTOR

. The term speed is used to denote the

distance traveled per unit time, and velocity is this

number along with an indication as to which direction

this motion occurs.

The rate of change of velocity is called accelera-

tion. Its magnitude is given by the first derivative of

velocity (and, consequently, the second derivative of

displacement). It too is considered a vector and is

assigned a direction.

Acceleration ais the first derivative of veloc-

ity and the second derivative of distance:

Physicists often follow S

IR

I

SAAC

N

EWTON

’s notation of

denoting differentiation with respect to time with a dot

and of denoting displacement with the letter s. We

have: a=·

v= ¨

s.

If the position of a moving object is given by a set

of

PARAMETRIC EQUATIONS

x= x(t) and y= y(t),then

the position vector of the object is the vector

<x(t),y(t)>, its velocity is the vector < ·

x(t),·

y(t)>, and its

acceleration is the vector < x¨ (t),ÿ(t)>.

Scientists at NASA sometimes use the term jerk to

denote the rate of change of acceleration. Astronauts

accelerating at a uniform rate will be pressed back into

their seats with a constant force, leading to a smooth

ride. Any change in the value of acceleration leads to

changes in force pressures.

Acceleration due to gravity, denoted g, is the accel-

eration with which an object falls freely to earth unim-

peded by air resistance. For many centuries it was

believed that more massive objects would fall faster

than lighter objects, but in 1638, G

ALILEO

G

ALILEI

(1564–1642) demonstrated, by theory and by experi-

ment, that this is not the case: all falling objects acceler-

ate at the same rate irrespective of their mass. (If

acceleration were dependent on mass, at what rate

would two falling bodies of different mass tied together

fall?) The accepted value for gis 9.80665 m/sec/sec, but

this magnitude varies at different locations of the Earth

due to the fact that the Earth is not a perfect sphere.

(At the poles its value is 9.8321 m/sec2and at the equa-

tor, 9.7799 m/sec2.)

Venn, John (1834–1923) British Logic, Probability

theory Born on August 4, 1834, in Hull, England,

logician John Venn is remembered for introducing and

popularizing the use of diagrams of overlapping circles

as a means to represent relations between sets. Although

such diagrams were used decades earlier by both G

OT

-

TFRIED

W

ILHELM

L

EIBNIZ

(1646–1716) and L

EONHARD