calculus and physics
posted by olivia .
An explosion causes debris to rise vertically with an initial velocity of 75 ft/sec.
a. In how many seconds does it attain maximum height?
b. What is the maximum height?
c. What is the speed of the debris as it reaches 12 ft on the way back to the ground?
d. What is the velocity of the debris at 20 ft on the way up

Easy way is to assume that total energy is constant (of course debris might have a lot of air drag but there is not much we can do about that.
mass m (will cancel everywhere in this problem)
(1/2) m v^2 + m g h = total energy
at start h = 0 and v = 75
so
(1/2)(75)^2 = total energy/m = 2813
in obsolete English units as here g = 32 ft/s^2
so at 12 ft
(1/2) v^2 = 32(12)
solve for v
and at 20 ft
(1/2) v^2 = 32 (20) solve for v
note that the speed is the same upwards and downwards at the same height, just the direction changes. 
at top
(1/2) v^2 = 0
so
32 h = 2813
at 12 feet
(1/2)v^2 + 32 (12) = 2813
at 20 feet
(1/2) v^2 + 32 (20) = 2813 
a)
Height(t) = 16t^2 + 75t
Height ' (t) = 32t + 75 = 0 for a max of Height
32t = 75
t = 75/32 seconds or appr 2.34 seconds
b) Height(75/32) = 16(75/32)^2 + 75(75/32) = 87.89 ft
c) when Height = 12
12 = 16t^2+ 75t
16t^2  75t + 12 = 0
t = 4.5216 sec (on the way down) or
t = .1659 seconds (on the way up)
velocity(t) = height ' (t)
= 32t + 75 , so we want it on the way down
velocity(4.5216) = 32(4.5216)+75 = 69.69 ft/sec
d)
same method as in c)
16t^2  75t + 20 = 0
t = 4.40 seconds (on the way down), or
t = .284 secpmds (on the way up
velocity(.284) = 32(.284)+75 = 65.92 ft/sec