Question

A ball is dropped from a height of 200 ft. and each time it strikes the ground it rebounds to a height of two-thirds of the distance from which it fell. find the total distance traveled by the ball before it comes to rest.

Answer 1

if you set it in that perspective then it will never get to rest because understanding the concept that if you take 2/3 of 200ft every time it strikes the ground and rebounds then the distance will just keep getting smaller and smaller and it will never reach zero. therefore giving you and infinity limit.

Answer 2

like for ex: half of 1 is .5, then half of .5 is .25, then half of .25 is .125 etc. So the value will just keep getting smaller but it will never reach zero

Answer 3

\[\huge \lim_{n->\infty} \left(\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^n 200\right) = 600\]

Answer 4

Exactly cyter, because it is saying that the distance will eventually get so small that we can neglect it. For ex: if the distance gets to .000000000000000000000000000025 then we can say it is basically zero

Answer 5

Notice that \[\huge \sum_{n=0}^\infty \left(\frac{2}{3}\right)^2\] is a geometric series, so the limit is;
\[\huge \frac{a_0}{1-\frac{2}{3}}\Rightarrow \frac { (\frac{2}{3})^0}{1-\frac{2}{3}} \Rightarrow \frac {1}{\frac{ 1}{3}} = 3 \]
\[\huge 3*200 = 600\]

Answer 6

the first series is the falling motion.
200 + 400/3 + 800/9 + 1600/18 +....

the second series the rising motion
400/3 + 800/9 + 1600/18

200 + 2(400)=1000.