# Determine how many times a ball has to bounce after which the height of the ball is 10% of the initial height.

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### 1 Answer

The problem relates to exponential decay. If a quantity decays exponentially, it decreases at a rate proportional to its value.

This is expressed as `(dH)/dt = -lambda*H` which gives `H(t) = Ho*e^(-lambda*t)`

Here, there is no particular value of the decay constant `lambda` that has been given.

After the ball bounces t number of times, its height is equal to `Ho*e^(-lambda*t)` where Ho is the initial height.

`0.1*Ho = Ho*e^(-lambda*t)`

=> `e^(-lambda*t) = 0.1`

take the natural log of both the sides

`-lambda*t = ln(0.1)`

=> `t = -ln(0.1)/lambda`

**The ball has to bounce** `t = -ln(0.1)/lambda` **times after which the height is 10% the original height if **`lambda` **is the decay constant.**

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