Question

A football is punted from a height of 8 feet above the ground with an initial vertical velocity of 44 feet per second. Write an equation to model the height h, in feet, of the ball t seconds after it has been punted. The football is caught at 15 feet above the ground. How long was the football in the air?

Answer 1

Assuming ballistic motion, the equation for height as a function of time is
h(t) = -16t² + v0·t + h0
where v0 is the initial vertical velocity (44 ft/s) and h0 is the initial height (8 ft).

You want to find t such that h(t) = 15. Substituting the given values, we have
15 = -16t² +44t +8
16t² -44t +7 = 0
Using the quadratic formula, we find t to be ...
t = (44 ±√(44² - 4·16·7))/(2·16)
t = (11 ± √93)/8

We are not interested in the time when the football is rising through the 15 ft height, so the time of interest is
t = (11 +√93)/8 ≈ 2.580 . . . . seconds