Using a powerful air gun, a steel ball is shot vertically upward with a velocity of 80 meters per second, followed by another shot after 5 seconds. Find the initial velocity of the second ball in order to meet the first ball 150 meters from the ground.

To solve this problem, we need to use the equations of motion.

Step 1: Determine the time it takes for the first ball to reach 150 meters.

We know that the height (h) is given by the equation h = ut - 0.5gt^2, where u is the initial velocity, g is the acceleration due to gravity (9.8 m/s^2), and t is the time.

So, 150 = 80t - 0.59.8t^2.

Solving this quadratic equation, we get two values for t: t = 3.08 seconds or t = 6.15 seconds. Since the ball cannot reach the height of 150 meters in a time greater than the total time of flight (which is less than 6.15 seconds), we discard the value of t = 6.15 seconds.

So, the first ball takes 3.08 seconds to reach a height of 150 meters.

Step 2: Determine the time the second ball has to reach 150 meters.

The second ball is shot 5 seconds after the first, so it has less time to reach the height of 150 meters. This time is 3.08 seconds - 5 seconds = -1.92 seconds. Since time cannot be negative, this means that the second ball must be shot with a higher initial velocity to reach 150 meters at the same time as the first ball.

Step 3: Determine the initial velocity of the second ball.

We use the same equation as in step 1, but this time we solve for u:

150 = u1.92 - 0.59.8*1.92^2.

Solving for u, we get u = 117.6 m/s.

So, the initial velocity of the second ball must be 117.6 m/s for it to meet the first ball 150 meters from the ground.