a ball is dropped from a height of 24m. With each bounce, the ball reaches a height that is half the height of the previous bounce. How far has the ball travel

Question

a ball is dropped from a height of 24m. With each bounce, the ball reaches a height that is half the height of the previous bounce. How far has the ball travelled by the time it comes to a rest( the answer has to be a whole number)

Answer 1

Total distance traveled = D (m)

 

D = 24 +(12 + 12) + (6 + 6) + (3 + 3) + (3/2 + 3/2) + ... = 24 + 2 (12 + 6 + 3 + 3/2 + ...)

 

= 24 +2(12 + 12/2 + 12/4 + 12/8 + ...) = 24 + 24[(1/2)⁰ + (1/2)¹ + (1/2)² + (1/2)³ + ...]

 

Note that the expression in brackets [ ] is a geometric series:

 

1 + x + x² + x³ + ...

 

where x = 1/2

 

If |x| < 1, then

 

1 + x + x² + x³ + ... = 1/(1-x)

 

Then the expression in brackets [ ] is

 

1/(1 - 1/2)  = 2

 

Finally, 

 

D = 24 + 24(2) = 24 + 48 = 72

 

D = 72 m