1. Model the ball’s height h (in feet)

Matt and his friends are enjoying an afternoon at a baseball game. A batter hits a towering home run, and Matt shouts, “Wow, that must have been 110 feet high!” The ball was 4 feet off the ground when the batter hit it, and the ball came off the bat traveling vertically at 80 feet per second.

1. Model the ball’s height h (in feet) at time t (in seconds) using the projectile motion

model h (t) = -16 t 2 + v 0 t + h 0 where v 0 is the projectile’s initial vertical velocity

(in feet per second) and h 0 is the projectile’s initial height (in feet). Use the model

to write an equation based on Matt’s claim, and then determine whether Matt’s

claim is correct.

1. Did the ball reach a height of 100 feet? Explain.
2. Let h max be the ball’s maximum height. By setting the projectile motion model

equal to h max , show how you can find h max using the discriminant of the quadratic

formula.

1. Find the time at which the ball reached its maximum height.

Solution

4. a) The ball’s height h at time t is given by h (t) = -16t 2 + 80t + 4. Matt’s claim

is that h (t) = 110 at some time t. Applying the discriminant of the quadratic

formula to the equation -16t 2 + 80t + 4 = 110, or -16t 2 + 80t – 106 = 0,

gives b 2 – 4ac = 80 2 – 4 (-16) (-106) = 6400 – 6784 = -384. Since the discriminant is

negative, there are no real values of t that solve the equation, so Matt’s claim is incorrect.

100. b) For the ball to reach of height of 100 feet, h (t) must equal 100. Applying the discriminant

of the quadratic formula to the equation -16t 2 + 80t + 4 = 100, or

-16t 2 + 80t – 96 = 0, gives b 2 – 4ac = 80 2 – 4 (-16) (-96) = 6400 – 6144 = 256.

Since the discriminant is positive, there are two real values of t that solve the equation,

so the ball did reach a height of 100 feet at two different times (once before reaching its

maximum height and once after).

1. Setting h (t) equal to h max gives -16 t 2 + 80t + 4 = h max = 0, or -16 t 2 + 80t + 4 – h max = 0.

Since the maximum height occurs for a single real value of t, the discriminant of the

quadratic equation must equal 0.

b 2 – 4ac = 0

80 2 – 4 (-16) (4 – h max ) = 0

6400 + 64 (4 – h max ) = 0

64 (4 – h max ) = -6400

4 – h max = -100

– h max = -104

h max = 104

So, the ball reached a maximum height of 104 feet.

1. Solve the equation -16t 2 + 80t + 4 = 104, or -16t 2 + 80t – 100 = 0, using the quadratic

formula. You already know that the discriminant is 0 when the ball reached its maximum

height, so t = -80 } √0―

________ 2 (-16)

= -80 ___ -32 = 2.5. So, the ball reached its maximum height 2.5 seconds

after it was hit.