CPPC 3/14/17
WS Parametric Equations
Name: _______________________
Show all work in algebraically solving each problem.
1)
The center field wall in a baseball park is 7 feet high and 400 feet from home plate.
David Freese hits the ball 3 feet above the ground. It leaves the bat with a direction
angle of 15˚at a speed of 146.67 feet per second (100 miles per hour).
a)
Write a set of parametric equations for the path of the baseball.
( )
= 3+146.67*T * sin (15) −16*t
x1 = 146.67*T * cos 15
!
b)
y1
2
Is the hit a home run?
When is x=400 then plug that time into y1 to find height. Height = -17.37. No HR
( )
400 = 146.67*T * cos 15
400
=T
146.67* cos 15
( )
2.82 = T
!
c)
(
)
y1 2.82 = −17.37
If you answered “no” to part b, for an outfielder to catch the ball how far from home plate
should he stand? Assume the best height for the outfielder to catch the ball is 6 feet.
If you answered “yes” to part b, how high would the outfielder need to jump to keep
the ball from going over the outfield wall? Is this possible?
Find T when ball is at 6 feet. Then plug into x1 to find distance from home.
−16 → a,146.67sin(15)→ b,−3→ c
−b
b2 − 4ac
±
2a
2a
1.186 ± −1.104
t = 0.082,t = 2.291
( )
x (2.291) = 324.53 feet
!
x1 0.082 = 11.60 feet
1
CPPC 3/14/17
Name: _______________________
WS Parametric Equations
2) Greg Zuerlein punts the ball at an angle of 55˚ to the ground. He punts the ball at a
speed of 65 feet per second. The ball leaves his foot at 4.8 feet above the ground.
a)
Write parametric equations modeling the position of the ball.
( )
x1 = 65*T * cos 55
!
b)
c)
( )
y1 = 4.8 + 65*T * sin 55 −16*t 2
If Zuerlein is punting the ball from the 45-yard line (55 yards (165ft) from the end
zone), will the ball land in the end zone (End Zone is 10 yards deep (195 ft))?
Either find when y=0 and then find distance
Find t when x is 165 and 195 and find height. Already below the ground before
endzone.
165 = 65*t * cos(55)
195 = 65*t * cos(55)
165
=t
195
=t
65cos(55)
65cos(55)
4.426 = t
5.23 = t
y1(4.426) = −72.9
!
How far from the punter should the punt return man stand in order to catch the ball?
Assume he will catch the ball 5 feet off the ground.
Find when height is 5 feet.
−16 → a,65sin(55)→ b,−.2→ c
−b
b2 − 4ac
±
2a
2a
1.664 ± −1.660
t = 0.004,t = 3.324
( )
x (3.324 ) = 123.93 feet
!
x1 0.004 = 0.14 feet
1
d)
What is the hang time (the length of time the ball is in the air) of the punt?
If the punt is caught then 3.324. If not solve for t when c=4.8
e)
What is the maximum height of the ball? At what time does this occur?
−b
= 1.664 is when it is at maximum height. Y1(1.664)=49.1 feet
!2a
CPPC 3/14/17
WS Parametric Equations
Name: _______________________
3)
Annika Sorenstam needs to hit a ball over a tree 10 meters high and she wants to hit the
green 130 meters from her position. The tree is 30 meters from the golfer and blocks the
view of the hole, which is on level ground with respect to the golfer. Assume you can
neglect air resistance.
a)
If Annika hits the ball with a velocity of 35 m/sec at angle of 52˚, what parametric
equations model the position of the ball?
( )
= 35*T * cos (52) − 4.9*t
x1 = 35*T * cos 52
!
b)
y1
2
Will Annika clear the tree? Yes it clears the tree.
( )
30 = 35*T * cos 52
30
=t
35cos(52)
1.39 = t
! y1(1.39) = 28.9
c)
Will she land the golf ball on the green? No below ground when x=130
( )
130 = 35*T * cos 52
130
=t
35cos(52)
6.03 = t
! y1(6.03) = −12.0
d)
Assume the hole is 130 meters from
Annika’s position. With what initial
velocity must Annika hit the ball to
get a hole in one? Assuming that
she is able to do this, what is the
tree that she can clear?
Change Vel. And redo problem b.
( )
y = v *T * sin 52 − 4.9*t 2
( )
130 = v *T * cos 52
130
=v
T * cos(52)
0=
( )
130
*T * sin 52 − 4.9*T 2
T * cos(52)
0 = 130tan(52)− 4.9T 2
( )
( )
−130tan 52 = −4.9T 2
tallest
−130tan 52
= T2
−4.9
( ) =T
−130tan 52
−4.9
5.83 = T
130
V=
= 36.23
5.83*
cos(52)
!
CPPC 3/14/17
Name: _______________________
WS Parametric Equations
4) Mia Hamm kicks a ball inside of mid-field approximately 75 feet from the goal. The
goal is 7 feet tall.
a)
She kicks the ball with an initial velocity of 54 feet per second at an angle of 36˚ with the
ground. Write parametric equations for the kick.
( )
= 52*T * cos (36) −16*t
x1 = 52*T * cos 36
!
b)
y1
2
If the goalie can block a ball up to 6 feet off the ground, will Mia score a goal? Will the
shot be blocked by the goalie or will the shot travel over the net? In your answer
explain the information you used to make your decision. In your explanation include
any times, horizontal distances, and vertical distances that helped you answer the
question.
52*T * cos 36 = 75
Find time when x=75 plug into y to find height
75
After 1.78 seconds the ball is at the goal and is 3.63 feet off the
T=
= 1.78
ground. Easily blocked
52cos(36)
! y(1.78) = 3.63
( )
c)
Assume that Mia kicked the ball with an initial velocity of 48 feet per second trying to
set up a teammate for a header. How far from Mia should her teammate stand if she
can score a goal if the ball is a distance of 6 to 7 feet off the ground? How long will the
ball be in the air?
( )
48*T * cos (36) −16*t
48*T * cos 36 −16*t 2 = 6
2
−6 = 0
( )
48*T * cos (36) −16*t
48*T * cos 36 −16*t 2 = 7
2
−7 = 0
−16 → a,48sin(36)→ b,−6 → c −16 → a,48sin(36)→ b,−7 → c
( )
= 48*T * cos (36) −16*t
x1 = 48*T * cos 36
!
y1
2
−b
b2 − 4ac
±
2a
2a
0.88 ± −0.63
t = 0.25,t = 1.52
( )
x (1.52) = 63.78 feet
!
x1 0.25 = 10.4 feet
1
−b
b2 − 4ac
±
2a
2a
0.88 ± −0.58
t = 0.30,t = 1.46
( )
x (1.52) = 61.6 feet
!
x1 0.30 = 12.6 feet
1
CPPC 3/14/17
WS Parametric Equations
Name: _______________________
5)
Mat Hoffman’s rear bicycle tire has a trademark (bright red dot) on its extreme outer
edge. The radius of the tire is 13.5 inches, and the trademark is initially located at the
3:00 o’clock position.
a)
Make a sketch of the wheel on a
rectangular coordinate system. Let the
horizontal axis represent the ground
and the vertical axis contain the center
of the wheel. Mark the origin O, the
trademark M, and the center of the
wheel C.
b)
Mat’s wheel makes 4 revolutions per second. Write parametric equations that model the
location of the trademark on the turning wheel at time t .
x = 13.5* cos(8π *t )
! y = 13.5* sin(8π *t )+13.5
c)
Provide a window that could be used to simulate the first revolution of the trademark.
−13.5 ≤ x ≤ 13.5
0 ≤ y ≤ 27
!0 ≤ t ≤ .25,ΔT = .01
d)
Adjust the equations so that the trademark starts at the 6:00 position.
π
x = 13.5* cos(8π *t − )
2
π
y = 13.5* sin(8π *t − )+13.5
2
!
e)
How far above the ground is the trademark at t = 2.3 ?
Y1(2.3)=9.33 inches
CPPC 3/14/17
WS Parametric Equations
Name: _______________________
6)
Mr. Rust’s tractor has Everwear tires, which have a small bright orange letter V painted
on the extreme outer edge. The radius of the tire is 2.5 feet and the initial position of the
V is 3:00 o’clock.
a)
Make a sketch of the tire on this
coordinate system. The vertical axis
should contain the center of the tire.
Mark the origin O, the orange V as V,
and the center of the tire C.
b)
Suppose the tire makes 2 revolutions per second. Write parametric equations that model
the location of the V on the tire at any time t .
x = 2.5* cos(4π *t )
! y = 2.5* sin(4π *t )+ 2.5
c)
Provide a window that could be used to simulate the first revolution of the V.
d)
Adjust the equations so that the V starts in the 12:00 position.
x = 2.5* cos(4π *t + π /2)
! y = 2.5* sin(4π *t + π /2)+ 2.5
e)
How high above the ground is the V at t = 1.9 ?
Y(1.9)=3.27