Linear Law - Predictions from Linearised Models
Linear Law - Predictions from Linearised Models
A particle moves through a medium with speed \(v\) ms\(^{-1}\) and experiences a resistance force \(F\) (in newtons). It is suggested that the relationship between \(F\) and \(v\) follows the equation:
\(F = kA^v\) where \(k\) and \(A\) are constants. The table below shows experimental values of the speed \(v\) and the corresponding force \(F\) :
| Speed, \(v\) ms\(^{-1}\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|
| Force, \(F\) newtons | \(14.0\) | \(19.6\) | \(27.4\) | \(38.4\) | \(53.8\) |
- Rewrite the equation \(F = kA^v\) in a form suitable for plotting a straight line graph.
- Using the data provided:
- Plot an appropriate straight line graph and use it to estimate the values of \(A\) and \(k\).
- Use your graph to find the value of the force \(F\) when the speed is \(3.5\) ms\(^{-1}\).
- Draw a suitable straight line on your graph to help solve the equation \(kA^v = 4^v\). Use your graph to estimate the value of \(v\) that satisfies the equation \(kA^v = 4^v\).
Solution:
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Answer:(a) \(\lg F=(\lg A)v+\lg k\). (b)(i) \(A\approx1.40,\ k\approx10\) (ii) \(F\approx32.5\text{ N}\). (c) Additional line \(\lg F=v\lg4\); \(v\approx2.2\text{ m/s}\).
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