Answer
The minimum time required to raise the bucket 12.0 meters is 2.58 seconds.
Work Step by Step
We can use a force equation to find the maximum possible acceleration of the bucket. We can set the tension force in the cord to $F_T = 75.0 N$.
$\sum F = ma$
$F_T - mg = ma$
$a = \frac{F_T-mg}{m} = \frac{(75.0~N) - (5.60~kg)(9.80~m/s^2)}{5.60~kg}$
$a = 3.593~m/s^2$
We can find the time to move up 12.0 meters.
$y = \frac{1}{2}at^2$
$t = \sqrt{\frac{2y}{a}} = \sqrt{\frac{(2)(12.0~m)}{3.593~m/s^2}}$
$t = 2.58~s$
The minimum time required to raise the bucket 12.0 meters is 2.58 seconds.
