Final answer:
To calculate the average speed of a rubber ball that bounces to 81/100 of its previous height after each bounce, one would have to sum the total distance covered by the ball as an infinite geometric series and then divide this total distance by the sum of the total time taken for each bounce, also an infinite series.
Step-by-step explanation:
To find the average speed of the rubber ball, we must consider its motion throughout the entire journey. We start by analyzing the ball's initial descent from 5 m. Using the formula v^2 = u^2 + 2gh, where v is the final velocity, u is the initial velocity (which is zero, as the ball starts from rest), g is the acceleration due to gravity (10 m/s^2), and h is the height (5 m), we find that the ball's speed just before the first bounce is sqrt(2gh). Then, the ball rises to 81/100 of its previous height after each bounce. The total distance traversed by the ball includes all the descents and ascents.
The total distance (S) traversed before it comes to rest is given by the sum of an infinite geometric series:
S = a + ar + ar^2 + ar^3 + ...
where a is the first term (5 m, in this case) and r is the common ratio (0.81).
The sum of this series is S = a / (1 - r). Plugging the numbers in, we get the total distance. The time taken for each bounce can be calculated using the formula t = 2v/g, for the descent and ascent individually, then, using a similar series formula to find the total time.
The average speed is then calculated by dividing the total distance by the total time. However, without the summation of the time series, we cannot calculate the specific numerical answer. Thus, we can only establish that the average speed would be given by the formula:
Average Speed = Total Distance / Total Time