Question

Samar serves a volleyball over a net at an initial height of 5 feet with a velocity of 100 feet per second. this can be represented by the function h(t) = −16t2 100t 5. if the other team misses, about how long does it take the ball to hit the ground?

Answer 1

6.3 seconds

Explanation:

The time can be found by solving the equation for the time that makes the height be zero.

Equation

The equation we want to solve is ...

h(t) = 0

-16t² +100t +5 = 0 . . . . . equation for t when height is zero

Solution

We can find the solution to this equation by "completing the square."

-16(t² -6.25t) +5 = 0 . . . . factor out the leading coefficient

-16(t² -6.25t +3.125²) +5 +16(3.125²) = 0 . . . . add and subtract 16(6.25/2)²

-16(t -3.125)² +161.25 = 0

(t -3.125)² = -161.25/-16 . . . . . . subtract 161.25 and divide by -16

t = 3.125 +√(161.25/16) ≈ 6.2996

It takes about 6.3 seconds for the ball to hit the ground.

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Additional comment

We have ignored the negative solution, which is irrelevant to this problem.

Note that this equation for ballistic motion assumes the serve was made straight upward. The ball would reach a height of over 160 feet, given the upward initial velocity of 100 ft/s.