Quadratic Applications Vertical Motion

1
Quadratic Applications---------------------------
----Vertical MotionProfit / Income

• By Jeffrey Bivin
• Lake Zurich High School
• jeff.bivin_at_lz95.org

Last Updated November 30, 2007
2
Vertical Motion

• Compares the height of an object with the time in
  flight.

g force of gravity 32ft/sec or
9.8 m/sec
vo initial velocity
ho initial height
Jeff Bivin -- LZHS
3
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• What is the maximum height of the ball?
• When will the ball reach the maximum height?
• When will the ball return to the ground?
• When will the ball be at a height of 250 feet?
• When will the ball be at a height of 400 feet?
• When will the ball be at a height of 50 feet?
• If the ball lands in a 20 foot deep pit, when
  will the ball hit the bottom of the pit?
• What will be the height of the ball in 3 seconds?
• How far from the building will the ball land?

Jeff Bivin -- LZHS
4
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• What is the maximum height of the ball?

We need to use
g 32 ft/s2
vo 80 ft/s
ho 200 ft
?
200 ft
Jeff Bivin -- LZHS
5
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.
300 ft.

• What is the maximum height of the ball?

Where is the maximum?
Find the vertex
Vertex is
Jeff Bivin -- LZHS
6
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• When will the ball reach the maximum height?

2.5 sec
Where is the maximum?
Find the vertex
Vertex is
Jeff Bivin -- LZHS
7
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• When will the ball return to the ground?

6.830 sec.
What is the height at the ground?
h(t) 0
200 ft
Get the decimal approximations
Jeff Bivin -- LZHS
8
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• When will the ball be at a height of 250 feet?

0.732 sec 4.268 sec.
250 feet
What height?
h(t) 250
?
200 ft
Get the decimal approximations
Jeff Bivin -- LZHS
9
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• When will the ball be at a height of 400 feet?

never
What height?
h(t) 400
so?
Wait, what was the maximum height?
Jeff Bivin -- LZHS
10
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• When will the ball be at a height of 50 feet?

6.453 sec.
What height?
h(t) 50
Get the decimal approximations
Jeff Bivin -- LZHS
11
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• If the ball lands in a 20 foot deep pit, when
  will the ball
  hit the bottom of the pit?

What height?
h(t) -20
200 ft
20 feet below the ground
Jeff Bivin -- LZHS
12
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• If the ball lands in a 20 foot deep pit, when
  will the ball hit the bottom of the pit?

6.972 sec.
What height?
h(t) -20
Get the decimal approximations
Jeff Bivin -- LZHS
13
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• What will be the height of the ball in 3 seconds?

296 ft.
What time?
t 3
Jeff Bivin -- LZHS
14
A ball is thrown into the air from the top of a
200 foot tall building with an initial upward
velocity of 80 ft/sec.

• How far from the building will the ball land?

Wait !!!!
This formula compares time with height, not
horizontal distance.
Answer
we dont know!
200 ft
?
Jeff Bivin -- LZHS
15
That was FUN!
Let's do one more!
Jeff Bivin -- LZHS
16
A diver dives off a 3 meter diving board into a
pool with an initial upward velocity of 3.5
m/sec.

• What is the maximum height of the diver?
• When will the diver reach his/her maximum height?
• When will the diver splash into the water?
• What will be the height of the diver in 1 second?

Jeff Bivin -- LZHS
17
A diver dives off a 3 meter diving board into a
pool with an initial upward velocity of 3.5
m/sec.

• What is the maximum height of the diver?
• When will the diver reach his/her maximum height?
• When will the diver splash into the water?
• What will be the height of the diver in 1 second?

3.625 meters
0.358 sec.
1.217 sec.
1.6 meters
Jeff Bivin -- LZHS
18
That was FUN!
Let's do one more!
Jeff Bivin -- LZHS
19
A taxi service operates between two airports
transporting 200 passengers a day. The charge is
15.00. The owner estimates that 10 passengers
will be lost for each 2 increase in the fare.
What charge would be most profitable for the
service? What is the maximum income?
VERTEX
Income Price ? Quantity
Define the variable
f(x) ( 15 2x ) ( 200 10x )
x number of 2 price increases
f(x) 3000 150x 400x 20x2
f(x) income
f(x) 20x2 250x 3000
f(6.25) 20(6.25)2 250(6.25) 3000
f(6.25) 3781.25
Vertex is
So, price (15 2x) (15 2(6.25)) 15
12.5 27.50
Maximum income f(x) 3781.25
Jeff Bivin -- LZHS
20
That was FUN!
Jeff Bivin -- LZHS