Final answer:
The initial speed of the bullet is 79.93 m/s and the total flight time until the bullet reaches the ground is 16.3265 seconds.
Step-by-step explanation:
(a) To calculate the initial speed of the bullet, we can use the fact that the maximum height is reached when the vertical component of the velocity is zero. At the maximum height, the final velocity in the vertical direction is zero and the acceleration due to gravity is -9.8 m/s². Using the equation vf = vi + at, we can solve for the initial vertical component of velocity:
0 = vi - 9.8 * t, where t is the time it takes to reach the maximum height.
Using the equation h = vit + 1/2 * a * t², we can solve for t:
256 = 0 + 1/2 * (-9.8) * t².
Solving for t gives us t = 8.16327 seconds
Substituting the value of t into the first equation, we can solve for the initial vertical component of velocity:
0 = vi - 9.8 * 8.16327
solving for vi gives us vi = 79.93 m/s.
(b) To find the total flight time, we can use the fact that the time it takes to reach the maximum height is half of the total flight time. So, the total flight time is 2 times the time it takes to reach the maximum height, which is 2 * 8.16327 seconds = 16.3265 seconds.