Parabolic Golf Shot Equations

Date: 01/24/2002 at 09:10:36
From: Michelle Doyon
Subject: Solving a quadratic word problem

One golfer hits a ball off a tee toward a hole that is 195 yd away. 
The hole is surrounded by a green with a radius of 10 yd. An equation 
of the path of the ball is y = x-0.005x^2, where x is the horizontal 
distance the ball travels and y is the height of the ball. A second 
golfer's drive can be modeled by the equation y = 1.5x-0.008x^2.
Does either player land the ball on the green?

I tried drawing a picture of the information I was given. The result 
makes it look as if the ball travels in an upside-down U (a negative 
parabola). The radius of the green is 10 yds. The whole green is 
20 yds. across. 

I don't know if I can use a^2 + b^2 = c^2. or if I use -b +- the 
sqaure root of b squared - 4(a)(c) all over 2(a). I tried because 
I thought that would get me where the ball started and where the ball 
landed, but I need help. 

Thank you,
Michelle

Date: 01/24/2002 at 09:36:08
From: Doctor Ian
Subject: Re: Solving a quadratic word problem

Hi Michelle,

You got the right shape for the curve. A parabola is the shape of the 
trajectory followed by a ballistic object (i.e., one that receives an 
initial impulse and is afterwards affected only by gravity, like a 
cannonball, or a bullet, or a golf ball).  

Here's how I would approach the problem. If you choose some value 
of x - say 100 yards - then you can find the height of the ball above 
the ground:

  y = 100 - (5/1000)(100^2)

    = 100 - 50

    = 50 yards above the ground

What about at 1000 yards?

  y = 1000 - (5/1000)(1000^2)

    = 1000 - 5000

    = -4000 yards 'above' the ground

Obviously, the ball isn't 4000 yards below the ground, but the minus 
sign tells us that the ball would hit the ground sometime before the 
value of x reaches 1000 yards. 

If the ball lands on the green, it will be true that the height of the 
ball at 185 yards (the closest point on the green) will be zero or 
positive; and the height of the ball at 205 yards (the farthest point 
on the green) will be zero or negative. Do you see why this is true? 

So one way to determine whether a ball lands on the green is to 
evaluate its height at both distances, and see if it changes sign. 

Another way, of course, is to set the height of the ball to zero,

  0 = x - (5/1000)x^2

and solve the quadratic equation for x. This should give you the 
distances (i.e., values of x) at which the ball is at ground level.  
(Note that one of them should be at the tee, i.e., x = 0, or you 
haven't solved the equation correctly.) 

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/