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