A Word from SpaceX
Average and Instantaneous Velocity
Average and Instantaneous Velocity
or
Your Car’s Speedometer Knows Calculus
Peter A. Perry
University of Kentucky
September 2, 2016
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
Bill of Fare
1. Who Cares About Calculus? A Word from SpaceX
2. Average and Instantaneous Velocity
3. Falling Bodies
4. Limits Preview
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
This is A Test of REEF
What is wrong with Dr. Perry’s clothing?
A His tie is not properly tied
B His shirt is not tucked in
C His glasses are too big
D His glasses are the wrong color
E None of the above
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
A Word from SpaceX
SpaceX, founded by former PayPal CEO Elon Musk, is a private
corporation devoted to developing reusable rockets and enabling
interplanetary travel. The Grasshopper rocket is a test vehicle, 106
feet tall equipped with a single Merlin-1 engine developing 155,000
pounds of thrust. It was used as a test vehicle to develop a control
system that would enable SpaceX rockets to land “with the
precision of a helicopter.”
Let’s watch the Grasshopper do its work here.
Let’s see what this control system can do on a real rocket here.
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
What is Velocity?
Let’s begin with a familar formula from Physics:
d = rt
(Distance is rate of speed times time). Let’s allow the “speed” to
have a sign, + for forward motion and − for backward motion.
Signed speed is called velocity because it has a direction as well as
a magnitude. Instead of r we’ll write v . From the equation
v=
d
t
so velocity is (signed) distance travelled divided by elapsed time.
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
What is Average Velocity?
1. Suppose I drive from Lexington to Louisville (distance 90
miles) in two hours. My average velocity is:
2. Suppose I go from mile point 110 on Interstate 75 at 2:00 PM
to mile point 185 on Interstate 75 at 2:45. What is my
velocity in miles per hour?
3. Suppose I drop a ball off the top of Patterson Office Tower
(180 feet high) and it lands at ground level 3 seconds later.
What was its average velocity during the trip downward,
measured in feet per second?
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
What is Average Velocity?
1. Suppose I drive from Lexington to Louisville (distance 90
miles) in two hours. My average velocity is:
45 miles per hour
2. Suppose I go from mile point 110 on Interstate 75 at 2:00 PM
to mile point 185 on Interstate 75 at 2:45. What is my
velocity in miles per hour?
Change in position = 185-110 = 75 miles; Change in time =
75 miles
miles
45 minutes = 3/4 hour, so v = 3/4
hour = 100 hour
3. Suppose I drop a ball off the top of Patterson Office Tower
(180 feet high) and it lands at ground level 3 seconds later.
What was its average velocity during the trip downward,
measured in feet per second? Change in position = 0-180 =
ft
-180 ft; change in time = 3 seconds, so v = −180
3 = −60 sec
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
What is Average Velocity?
If a body goes from distance d1 at time t1 to distance d2 at time
t2 from a chosen starting point, its average velocity during the
time interval from t1 to t2 is
vav =
d2 − d1
t2 − t1
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
CSI
Professor Perry travels from Lexington to Cincinnati in 2 hours, a
distance of 90 miles, but receives a notice from the State Police
that he was clocked at 120 miles per hour near Florence, Kentucky.
At his trial, he tells the judge that this is impossible.
“Your honor, I am a mathematician. I left my home at 2:00 PM
and arrived in Cincinnati at 4:00 PM. My home is 90 miles from
Cincinnati. This means my average speed was only 45 miles per
hour, so I couldn’t have been driving 120 miles per hour while I
was on the highway. It’s mathematically impossible!”
If you were the judge, would you believe this? Why or why not?
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
CSI, Continued
The arresting policeman presents the following table of data to the
judge:
Time
HH:MM
3:15
3:16
3:17
Highway
Milepoint
156
157
159
Help the criminal justice system convict this outrageous liar by
computing Perry’s average speed:
• Between 3:15 and 3:16
• Between 3:16 and 3:17
Be careful to compute your answer in miles per hour!
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
CSI, Continued
The arresting policeman presents the following table of data to the
judge:
Time
HH:MM
3:15
3:16
3:17
Highway
Milepoint
156
157
159
Help the criminal justice system convict this outrageous liar by
computing Perry’s average speed:
• Between 3:15 and 3:16 60 mph
• Between 3:16 and 3:17 120 mph
Be careful to compute your answer in miles per hour!
A Word from SpaceX
Average and Instantaneous Velocity
This is a Test of REEF Polling
Ladies and Gentlemen of the Jury, How do you Find?
A I vote to convict
B I vote to acquit
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
What is Instantaneous Velocity?
Instantaneous velocity is the instantaneous rate of change of
distance with respect to time.
If you’re driving a car down the highway, instantaneous velocity is
what your speedometer tells you!
A Word from SpaceX
Average and Instantaneous Velocity
How do we calculate instantaneous velocity?
Since this is a calculus course, we want a method to calculate
instantaneous speed given a function that computes distance
travelled as a function of time. First of all we need an idea.
Average velocity gives a good approximation to instantaneous velocity if we measure average velocity over small time intervals. The smaller the time interval,
the better the approximation to instantaneous velocity.
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
How do we calculate instantaneous velocity?
Here’s some more data on the scofflaw college professor.
Time
HH:MM:SS
3:16:00
3:16:01
3:16:02
3:16:03
Highway
Milepoint
157.0
157.05
157.07
157.1
To approximate the speed of Professor Perry’s car at exactly 3:15
PM, compute the average speed in the following intervals:
• Between 3:16:00 and 3:16:03 (3 seconds)
• Between 3:16:00 and 3:16:02 (2 seconds)
• Between 3:16:00 and 3:16:01 (1 second)
Which gives the best approximation to the instantaneous speed at
3:16:00 ?
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
How do we calculate instantaneous velocity?
Here’s some more data on the scofflaw college professor.
Time
HH:MM:SS
3:16:00
3:16:01
3:16:02
3:16:03
Highway
Milepoint
157.0
157.05
157.07
157.1
To approximate the speed of Professor Perry’s car at exactly 3:15
PM, compute the average speed in the following intervals:
• Between 3:16:00 and 3:16:03 (3 seconds) 120 mph
• Between 3:16:00 and 3:16:02 (2 seconds) 126 mph
• Between 3:16:00 and 3:16:01 (1 second) 180 mph
Which gives the best approximation to the instantaneous speed at
3:16:00 ?
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
Instantaneous Velocity
Let’s take a look at this process of computing instantaneous
velocity by revisiting uniformly accelerated motion.
A bored tourist at the Grand Canyon throws a stone straight up
over the abyss with an initial speed of 64 ft/sec. Its height above
the ground in feet as a function of time in seconds is:
g (t ) = −16t 2 + 64t
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
Instantaneous Velocity
The stone’s height above the ground in feet as a function of time
in seconds is:
g (t ) = −16t 2 + 64t
Let’s try to find the instantaneous speed after 1 second. Using the
formula we compute
t
1 sec
1.0001 sec
1.001 sec
1.01 sec
1.05 sec
1.1 sec
g (t )
48 ft
48.00319984 ft
48.031984 ft
48.3184 ft
49.5600 ft
51.0400 ft
Find the average velocities on the intervals [1, 1.1], [1, 1.05],
[1, 1.01], [1, 1.001].
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
Instantaneous Velocity
The stone’s height above the ground in feet as a function of time
in seconds is:
g (t ) = −16t 2 + 64t
The average speeds are:
Interval
Speed
Average
[1, 1.1]
[1, 1.05]
[1, 1.01]
[1, 1.001]
[1, 1.0001]
30.4 ft/sec
31.2 ft/sec
31.84 ft/sec
31.984 ft/sec
31.9984 ft/sec
What’s your best guess for the instantaneous velocity at t = 1?
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
There Must Be a Better Way...
The stone’s height above the ground in feet as a function of time
in seconds is:
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + 0.1 is
g (1.1) − g (1)
0.1
The average speed between t = 1 and t = 1 + 0.01 is
g (1.01) − g (1)
0.01
The average speed between t = 1 and t = 1 + 0.001 is
g (1.001) − g (1)
0.001
The average speed between t = 1 and t = 1 + h is
???????????
A Word from SpaceX
Average and Instantaneous Velocity
Falling Bodies
There Must Be a Better Way...
The stone’s height above the ground in feet as a function of time
in seconds is:
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + 0.1 is
g (1.1) − g (1)
0.1
The average speed between t = 1 and t = 1 + 0.01 is
g (1.01) − g (1)
0.01
The average speed between t = 1 and t = 1 + 0.001 is
g (1.001) − g (1)
0.001
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
A Word from SpaceX
Average and Instantaneous Velocity
There Must be a Better Way
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
This is calculus, so let’s calculate!
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There Must be a Better Way
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
This is calculus, so let’s calculate!
g (1 + h ) = −16(1 + h )2 + 32(1 + h )
= 48 + 32h + 16h2
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There Must be a Better Way
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
This is calculus, so let’s calculate!
g (1 + h ) = −16(1 + h )2 + 32(1 + h )
= 48 + 32h + 16h2
g (1) = 48
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There Must be a Better Way
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
This is calculus, so let’s calculate!
g (1 + h ) = −16(1 + h )2 + 32(1 + h )
= 48 + 32h + 16h2
g (1) = 48
g (1 + h ) − g (1) = −16h2 + 32h
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There Must be a Better Way
g (t ) = −16t 2 + 64t
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
This is calculus, so let’s calculate!
g (1 + h ) = −16(1 + h )2 + 32(1 + h )
= 48 + 32h + 16h2
g (1) = 48
g (1 + h ) − g (1) = −16h2 + 32h
g (1 + h ) − g (1)
= −16h + 32
h
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There is, in fact, a better way!
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
g (1 + h ) − g (1)
= −16h + 32
h
The instantaneous speed is the number that the average speed
approaches as h → 0.
Falling Bodies
A Word from SpaceX
Average and Instantaneous Velocity
There is, in fact, a better way!
The average speed between t = 1 and t = 1 + h is
g (1 + h ) − g (1)
h
g (1 + h ) − g (1)
= −16h + 32
h
h
0.1
0.01
0.001
0
Average speed
on [1, 1 + h ]
−16(0.1) + 32 = 30.4
−16(0.01) + 32 = 31.84
−16(0.001) + 32 = 31.984
32
Falling Bodies