Drop Height Explain

Question

A ball is dropped from a certain height. The table shows the rebound heights of the ball after a series of bounces.

Bounce number , Height in centimeters

1 , 30

2 , 6

3 , 1

4 , 0

From what height, approximately, do you think the ball was dropped? Explain your reasoning.

Answer 1

Assuming an exponential decay to the bounces:

h(n) = h₀·bⁿ

1. h(n) = the ball's height on bounce n
2. b = the decay factor (unknown)
3. n = bounce number, starting at n = 0

To find b, take the ratio of successive terms:

6/30 = 0.2

1/6 = 0.167

0/1 = 0

It's not exact, they should all be the same, but they are close. Let's guess b = 0.2. That means that the ratio of the first bounce to the starting height should also be 0.2::

30/h₀ = 0.2

30/0.2 = h₀

150 cm = h₀

Now the equation is:

h(n) = 150·(0..2)ⁿ

Let's construct a table using the formula to see if approximates the table in the problem:

n h(n) height in problem

0 150 -

1 30 30

2 6 6

3 1.2 1

4 0.24 0

The exponential model gives close results. So 150 cm is a good estimate for the starting height.