(Solved): The log wind profile is a formula for determining wind speeds at different heights near the surface ...

The log wind profile is a formula for determining wind speeds at different heights near the surface of the Earth. Over open agricultural land with few buildings and obstructions, it is expressed as \[ v=v_{0} \frac{\ln (h / k)}{\ln \left(h_{0} / k\right)}, \] where \( v \) (in \( \mathrm{m} / \mathrm{s} \) ) is the wind speed at height \( h \) (in \( \mathrm{m} \) ) and \( v_{0} \) is the known wind speed at reference height \( h_{0} \). Meteorologists believe this relationship is accurate for heights up to \( 200 \mathrm{~m} \). The constant \( k \) is called a surface roughness factor. On different surfaces, this factor takes on different values. Over rough uneven terrain, \( k \) can be chosen to be \( 0.4 \). With \( v_{0}=10 \mathrm{~m} / \mathrm{s} \) at \( h_{0}=10 \mathrm{~m} \), determine the exact value of \( v \) and \( \frac{d v}{d h} \) at \( h=60 \mathrm{~m} \). Explain the meaning of your answers in practical terms.