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/
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