Question

Need help with this word problem, how am i supposed to know when it's at rest? wouldn't that be an infinite series?

suppose that a superball dropped from a height of 28 ft always rebounds 1/4 of the distance fallen. approximately how far does the ball travel before coming to rest?

Answer 1

The ball falls from 28 feet, bounces up to 7 feet, falls 7 feet, bounces up to 7/4 feet, falls 7/4 feet, .....
Find the sum of an infinite geometric series with common ratio r = 1/4 and the first term a = 7. Double the sum because it has to fall the same distance it has risen. In the end add the initial 28 feet to get the total distance traveled by the ball.

Answer 2

Why wouldn't the first term be 28?

Answer 3

In which case the \[S \infty = 37\]

Answer 4

The 28 is set aside and added at the end because the ball is initially dropped from 28 feet not thrown upwards to 28 feet and then falls by 28 feet and so the distance does not double. But after the first bounce the ball rises to 7' falls 7', rises to 7/4' and falls 7/4', and the distance doubles.

Sum of an infinite geometric series is:
a / (1 - r) = 7 / (1 - 1/4) = 7 / (3/4) = 7*4/3 = 28/3
Double the sum and add 28: 2*28/3 + 28 = 56/3 + 28 = 140/3 or 46.67 feet.

Answer 5

alternative
use this shortcut :

S = h (b+a)/(b-a)
S = 28 (4+1)(4-1) = 28 * 5/3 = 140/3 = 46.67

Answer 6

see the number 1/4 (rebounds "1/4" of the distance fallen)
b = 4 and a = 1

Answer 7

Yeah I saw that but what is the vocabulary for the variables a and b? If i wanted to look up the definition/explanation of the formula you just gave me what words would I be looking for? :)

r=ratio
\[a _{1}\]means first term
etc.

Is my question clear? I'm not sure if i'm asking it right...

Answer 8

it just a shortcut. i didnt use the formula of the sum of an infinite geometric series. my experience says that formula is work, but if the problem says if the ball rebounds a/b of the distance fallen till the ball is stop

Answer 9

Where did you learn that shortcut? It doesn't have a name? lol

Answer 10

im a teacher, i know how to using the right formula (but it already solved by @ranga above). so i gave you an alternative solution :)

Answer 11

So you figured that little shortcut out by yourself? and it doesn't have a name? and you haven't published it yet?

Answer 12

can you show me how to work to that formula? ... like how'd you figure it out

Answer 13

about that formula, this can be a long talking.. i ever show how that formula is born, you can read my profil (in answered questions section). i didnt remember when, may be 1 years ago :)

Answer 14

Any chance you'd remember the question? lol you have 3600 of them answered

Answer 15

hehehe,
well, little steps to get that formula
see this ilustration below :