Time The ball in Exercise 75 takes the following times for each fall. when when when
Chapter 9, Problem 76(choose chapter or problem)
Time The ball in Exercise 75 takes the following times for each fall.
\(\begin{array}{ll} s_{1}=-16 t^{2}+16, & s_{1}=0 \text { when } t=1 \\s_{2}=-16 t^{2}+16(0.81), & s_{2}=0 \text { when } t=0.9 \\s_{3}=-16 t^{2}+16(0.81)^{2}, & s_{3}=0 \text { when } t=(0.9)^{2} \\s_{4}=-16 t^{2}+16(0.81)^{3}, & s_{4}=0 \text { when } t=(0.9)^{3} \\\vdots & \vdots \\s_{n}=-16 t^{2}+16(0.81)^{n-1}, & s_{n}=0 \text { when } t=(0.9)^{n-1}\end{array}\)
Beginning with \(s_2\) the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by
\(t=1+2 \sum_{n=1}^{\infty}(0.9)^{n}\)
Find this total time.
Text Transcription:
s_1=-16t^2+16, s_1=0 when t=1
s_2=-16t^2+16(0.81), s_2=0 when t=0.9
s_3=-16t^2+16(0.81)^2, s_3=0 when t=(0.9)^2
s_4=-16t^2+16(0.81)^3, s_4=0 when t=(0.9)^3
s_n=-16t^2+16(0.81)^n-3, s_n=0 when t=(0.9)^n-3
s_2
t=1+2 sum_n=1^infinity (0.9)^n
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