Final answer:
The ball drops 640 meters in the next 8 seconds after the initial 4 seconds, based on the principle that the distance fallen is directly proportional to the square of time.
Step-by-step explanation:
Free Fall Distance Calculation
Since the distance d is directly proportional to the square of time t, we can say that d = k * t^2, where k is the constant of proportionality. Knowing that the ball drops 80 meters in the first 4 seconds, we can find the constant k using the equation 80 = k * 4^2, which simplifies to 80 = 16k or k = 5.
Now, to find out how far the ball drops in the next 8 seconds (t = 4 + 8 = 12 seconds), we sum up the distances for the first 4 seconds and the next 8 seconds separately, d1 = 5 * 4^2 and d2 = 5 * 12^2.
Finally, we calculate d2 - d1 to find the distance dropped in just the next 8 seconds.
To calculate the actual distances: d1 = 5 * 4^2 = 5 * 16 = 80 meters (which we already know), and d2 = 5 * 12^2 = 5 * 144 = 720 meters. Therefore, the ball drops 720 meters - 80 meters = 640 meters in the next 8 seconds.
This result is obtained by subtractive the initial 80 meters dropped from the total distance at 12 seconds, which gives us just the distance dropped during the 8 seconds interval.