3.4 Velocity, Speed, and Rates of Change

Distance =Displacement =

Change in position is measured in both

displacement distanceand

If you were to drive 10 miles east and then 4 miles west

10 miles east

4 miles west

The difference between the starting and ending positions of an object

The total amount of “ground covered” by an object or the total length of its path

14 miles6 miles

x

y

position

time

Distance =Displacement =

And the position graph would look like this:

10 miles east 4 miles west

14 miles6 miles

Consider a graph of position (distance from a starting point) vs. time.

time (hours)

position(miles)

Average velocity can be found by taking:

change in position

change in time

s

t

t

sA

B

ave

f t t f tsV

t t

The speedometer in your car does not measure average velocity, but instantaneous velocity.

0

limt

f t t f tdsV t

dt t

(The velocity at one moment in time.)

Velocity is the first derivative of position.

Mr. Murphy’s fall from the 196 foot platform canbe expressed with the equation:

216196)( tts …where t is in seconds and s is measured in feet. Given the above statement, the equation for his velocity is: ttv 32)( …which is expressed in what units?

feet/second

Find

Which is easy enough except…32 what?

)(tv 32)( tv

t (seconds)

v(t)

(fe

et/s

econ

d)

First, let’s look at the graph of velocity. Note that like the position graph, s(t), the y-axis represents velocity while the x axis represents time

Accelerationsec

sec/feet

2sec/32)( feetta feet/second2

Since the slope of a line is based on:x

y

…or in this case:

€

sΔt

=feet

sec

…we are now talking about:

€

vΔt

=feet /sec

sec

Is an expression of…

which is a rate of change of velocity.

Acceleration

Acceleration is the derivative of velocity.

dva

dt

2

2

d s

dt

€

a(t) = −32 feet /sec2 feet/second2

Remember: If you ever get lost, what can always save you?

UNITS!

Example: Free Fall Equation

21

2s g t

Speed is the absolute value of velocity.

GravitationalConstants:

2sec32

ftg

2sec8.9m

g

2sec980

cmg

2)32(2

1ts

216 ts

tdt

dsV 32

Wait! So what is the difference between speed and velocity?

Speed is the absolute value of velocity.

x

y

0v

If the object is moving upward…

…or to the right

0v

If the object is moving downward…

…or to the left

But speed does not indicate direction so speed will always be positive…

Wait! So what is the difference between speed and velocity?

Speed is the absolute value of velocity.

x

y

€

v = 40 ft /sec

If an object is moving upward at a speed of 40 ft/sec…

€

v = −40 ft /sec

If an object is moving downward at a speed of 40 ft/sec

But regardless of the direction, the speed will always be positive…

time

position

acc posvel pos &increasing

acc zerovel pos &constant

acc negvel pos &decreasing

velocityzero

acc negvel neg &decreasing acc zero

vel neg &constant

acc posvel neg &increasing

acc zero,velocity zero

It is important to understand the relationship between a position graph, velocity and acceleration:

WAIT! How can you tell that acceleration is positive? The slope (velocity) is tilting upward so the slope (velocity) is increasing

time

acc posvel pos &increasing

acc zerovel pos &constant

acc negvel pos &decreasing

velocityzero

acc negvel neg &decreasing acc zero

vel neg &constant

acc posvel neg &increasing

acc zero,velocity zero

Based on this graph, how can you use velocity and acceleration to determine when an object is speeding up or slowing down?

time

position

time

acc posvel pos &increasing

acc zerovel pos &constant

acc negvel pos &decreasing

velocityzero

acc negvel neg &decreasing acc zero

vel neg &constant

acc posvel neg &increasing

acc zero,velocity zero

0

0

a

v

0

0

a

v

0

0

a

v

0

0

a

vSpeeding

up

SlowingDown

time

position

v and a have the same sign

v and a have opposite signs

So the moral of the story is…

0

0

a

v

0

0

a

v0

0

a

v

0

0

a

vSpeeding

up

SlowingDown

Rates of Change:

Average rate of change = f x h f x

h

Instantaneous rate of change = 0

limh

f x h f xf x

h

These definitions are true for any function.

( x does not have to represent time. )