What's maximum height, how long, and until it reaches bottom

Question

what is the maximum height the ball reaches? how long will it take the baseball to reach the maximum height? how long does it take for it to reach the ground?

-16t^2+100t+25

Answer 1

What's maximum height, how long, and until it reaches bottom

what is the maximum height the ball reaches? how long will it take the baseball to reach the maximum height? how long does it take for it to reach the ground?

-16t^2+100t+25

h(t) = -16t² +100t + 25 ft

Ball reaches its maximum height when h'(t) = 0.

h'(t) = -16•2t + 100 = -32t + 100

-32t + 100 = 0

32t = 100

t = 3.125 s

h_max = -16•(3.125)² + 100•3.125 + 25

h_max = 181.25 ft

Ball reaches ground when h(t) = 0.

h(t) = 0 = -16t² +100t + 25

Factoring the quadratic by completing the square...

-16t² +100t + 25 = 0

t² − (100/16)t − (25/16) = 0

t² − (100/16)t − (25/16) = 0

t² − (25/4)t − (25/16) = 0

t² − (25/4)t = (25/16)

t² − (25/4)t + (−25/8)² = (25/16) + (625/64)

[t − (25/8)]² = (100/64) + (625/64)

√{[t − (25/8)]²}= √(725/64)

t − (25/8) = (5/8)•√29

t = (25/8) + (5/8)•√29

t ≈ 6.49 s

Answer 2

Hi there, Sam!

This problem is one that comes up in both Calculus and Physics all the time. You have a ball in parabolic flight. When it reaches the top of its flight it will change direction and start to fall. In other words, velocity is changing sign. At the moment of direction change, velocity is zero.

So you have

h = height = -16t^2 + 100t + 25 (and t is time from I assume the crack of the bat :-) )

hdot = velocity = dh/dt = -32t + 100

hdot = 0 at t = 3.125 sec. <--

Now, the height at that point is your maximum. That's -156.25 + 312.5 + 25 = 181.25 <--

You are going from there back to the ground. So how much time does it take to get h=0? Just for a sanity check, you know it's more than 3.125 sec, because that's how long it took to reach the top of the flight!

0 = -16t^2 + 100t + 25

t for the end of flight is (25 + 5 sqrt (29)) / 8. That's just over 6 seconds - about 6.3.

That answer I got using http://wolframalpha.com - highly useful site! But you may have a hand calculator that can do it. Good luck now!