Question

An engineer has developed a ball. He throws it from a
height of 48. The ball bounces to one half of the
previous bounce. He catches the ball after it has
travelled a total distance of 126. Find the number of
times the ball has bounced.​

Answer 1

Answer:

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