To calculate the sequence for the distance the ball travels as it falls and rises up the fourth time, we can break down the motion into individual phases: the fall phase and the rise phase. Let's denote the distance traveled in each phase as follows:
- Distance fallen in each phase: D_fall
- Distance risen in each phase: D_rise
Given that the ball bounces r% of the height h, we can determine the distances traveled in each phase as follows:
Fall Phase:
- The ball falls from height h, so the distance fallen in the first phase is D_fall = h.
Rise Phase:
- After the first fall, the ball rebounds to a height of r% of the previous height. Thus, the height reached after the first fall is h_rise = h * (r/100).
- In the rise phase, the ball falls from the height reached after the previous fall (h_rise) and rises back to that height. Therefore, the distance risen in each phase is D_rise = 2 * h_rise.
Now, let's calculate the distances traveled in each phase for the first four cycles:
First Cycle:
- Fall Phase: D_fall = h
- Rise Phase: D_rise = 2 * h_rise = 2 * (h * (r/100))
Second Cycle:
- Fall Phase: D_fall = h
- Rise Phase: D_rise = 2 * h_rise = 2 * (h * (r/100))^2 (since the ball rises from h_rise to h_rise)
Third Cycle:
- Fall Phase: D_fall = h
- Rise Phase: D_rise = 2 * h_rise = 2 * (h * (r/100))^3
Fourth Cycle:
- Fall Phase: D_fall = h
- Rise Phase: D_rise = 2 * h_rise = 2 * (h * (r/100))^4
The total distance traveled after each cycle is the sum of the fall and rise distances. Therefore, the sequence for the total distance traveled as the ball falls and rises up the fourth time is:
First Cycle: D_total = D_fall + D_rise = h + 2 * (h * (r/100)) Second Cycle: D_total = h + 2 * (h * (r/100))^2 Third Cycle: D_total = h + 2 * (h * (r/100))^3 Fourth Cycle: D_total = h + 2 * (h * (r/100))^4
Please note that this sequence assumes idealized conditions and neglects factors like air resistance and energy loss, which can affect the actual distances traveled.