My Favourite Problem No.2 Solution
Let’s start by calling the height the ball
was dropped from h.
h
After the first bounce we know it
reached 75% of this height, so we can
write this as 0.75h.
0.75h
After the next bounce it reached 75% of its height after the first bounce.
This is a height of
0.75 x 0.75h = 0.5625h
(which is 56.25% of the original height).
We are asked to find the number of bounces it takes for the height of the ball to be less than 10%
of the original height. In other words less than 0.1h so we need to keep finding 75% of the height
reached on the previous bounce.
After 3 bounces, the ball reaches 0.75 x 0.5625h = 0.4219h.
After 4 bounces, the ball reaches 0.75 x 0.4219h = 0.3164h.
After 5 bounces, the ball reaches 0.75 x 0.3164h = 0.2373h.
After 6 bounces, the ball reaches 0.75 x 0.2373h = 0.1780h
After 7 bounces, the ball reaches 0.75 x 0.1780 =0.1335h
After 8 bounces the ball reaches 0.75 x 0.1335 = 0.1001h
(close but still too high)
After 9 bounces the ball reaches 0.75 x 0.1001 = 0.0751h < 0.1h
After 9 bounces the ball’s rebound height is 7.51% of the original height.
The next part asks us to find the number of bounces before the ball rebounds less than 1% of the
original height. We could continue to multiply by 0.75 successively until the rebound height is less
than 0.01h, but is there an easier way to do it?
We can write the height of any bounce in terms of the height the ball was originally dropped from.
Notice that:
After 1 bounce,
After 2 bounces,
After 3 bounces,
the rebound height is 0.75h.
the rebound height is 0.75 x 0.75h = 0.752h
the rebound height is 0.75 x 0.75 x 0.75h = 0.753h and so on.
Therefore after 9 bounces,
the rebound height is 0.759h = 0.0751h
We know that 1% is 0.01 so we could try increasing powers of 0.75 until we find the solution. A
spreadsheet is a useful tool for doing this.
Power of 0.75
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
We see from the table that 0.7516 = 0.0100. This looks correct.
Value
0.7500
0.5625
0.4219
0.3164
0.2373
0.1780
0.1335
0.1001
0.0751
0.0563
0.0422
0.0317
0.0238
0.0178
0.0134
0.0100
0.0075
However 0.7516 > 0.0100 if you calculate it accurately.
Therefore after 17 bounces the ball rebounds to less than 1% of
the original drop height.
A graph of these values for increasing powers of 0.75 shows a distinct pattern. This is known as
exponential decay.
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
Power of 0.75
Further Investigation
The pattern you can see from looking at the height of the successive bounces is called a
geometric sequence.
You can express hn , the height of the ball after n bounces, as an equation like this:
hn = 0.75nh
To find the number of bounces when hn <
0.01h
you need to solve the equation
0.75nh = 0.01h
Divide both sides by h, gives
0.75n = 0.01
This type of equation is called an exponential equation. To solve this equation to find the value of n
you could use a spreadsheet or calculator and try different values.
A more direct way to solve this equation uses logarithms.
You might like to investigate these topics further by visiting:
http://www.mathsisfun.com/algebra/sequences-sums-geometric.html
http://www.mathsisfun.com/algebra/logarithms.html