# Question Video: Finding the Height of a Bouncing Ball Using Compound Decay Mathematics

A child’s ball loses 15% of its energy every time it rebounds. By considering that the ball’s kinetic energy is proportional to the height from which it was dropped, determine the height, to the nearest centimeter, that the ball must be dropped from so that it rebounds to 20 cm on the fifth bounce.

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### Video Transcript

A child’s ball loses 15 percent of its energy every time it rebounds. By considering that the ball’s kinetic energy is proportional to the height from which it was dropped, determine the height to the nearest centimeter that the ball must be dropped from so that it rebounds to 20 centimeters on the fifth bounce.

So, we know that the ball loses 15 percent of its energy every time it rebounds. But we’re also told that the kinetic energy is proportional to the height from which it was dropped. This means we can assume that the ball loses 15 percent of its height every time it rebounds. So, we want to find the original height that the ball must be dropped from so that the final bounce, or the fifth bounce in this case, is 20 centimeters high. Let’s define then that height to be equal to ℎ centimeters.

We might consider what happens, for instance, on bounce one. We know that the ball loses 15 percent of its height, so we could work out 15 percent of ℎ and subtract it. Alternatively, if we subtract 15 percent from 100 percent, we can see that each bounce must be equivalent to 85 percent of the height of the previous bounce. Dividing that by 100 and we can get a decimal multiplier, that is, 0.85. This means we can find the height of bounce one in terms of ℎ by multiplying the original height by 0.85. Bounce one then must be 0.85 times ℎ.

In a similar way, we can find the height of bounce two by multiplying the height of bounce one by 0.85 again. This can be alternatively written as 0.85 squared times ℎ. And we now might be spotting a pattern. If we continue in this way, bounce three will be equivalent to 0.85 cubed times ℎ centimeters. This means that bounce five will be equivalent in centimeters to 0.85 to the fifth power times ℎ. This is a really good example of a compound-decay formula.

So, with this in mind, we have an expression for the height of bounce five in terms of ℎ. But we know we want that to be equal to 20 centimeters. So, we can form an equation; that is, 0.85 to the fifth power times ℎ equals 20. We can then solve this for ℎ by dividing both sides by 0.85 to the fifth power. So ℎ is 20 divided by 0.85 to the fifth power, which gives us 45.0749 and so on. Correct to the nearest centimeter, that’s 45. So, the ball must be dropped from a height of 45 centimeters so that it rebounds to 20 centimeters on the fifth bounce.

Now, of course, we could actually check our answer. We can either do so by checking the height of the ball at every single bounce, in other words, substituting ℎ equals 45 into the first bounce then multiplying that by 0.85 consecutively. Or we can go ahead and substitute it straight into our expression for the height of bounce five, 0.85 to the fifth power times 45. That gives us 19.96, which correct the nearest centimeter is 20 centimeters. So, the ball must be dropped from a height of 45 centimeters.