The terminal velocity of a 4×10−5 {\rm kg} raindrop is about 9 {\rm m/s}. Assuming a drag force F_{\rm{D}} = - bv, Assuming a drag force determine the value of the constant b.
To determine the value of the constant b, we can use the formula for terminal velocity:
\(F_{\rm{D}} = -bv\)
where:
\(F_{\rm{D}}\) is the drag force,
\(b\) is the constant we need to find, and
\(v\) is the terminal velocity (9 m/s).
Since the raindrop has reached its terminal velocity, the net force acting on it is zero, meaning the drag force is equal in magnitude and opposite in direction to the gravitational force:
\(F_{\rm{D}} = F_{\rm{G}}\)
The gravitational force acting on the raindrop is given by the equation:
\(F_{\rm{G}} = mg\)
where:
\(m\) is the mass of the raindrop (4×10^−5 kg) and
\(g\) is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting these values into the equation, we have:
\(-bv = mg\)
Rearranging the equation, we can solve for \(b\):
\(b = \frac{{mg}}{{-v}}\)
Plugging in the values, we get:
\(b = \frac{{(4×10^−5 \, {\rm{kg}}) \cdot (9.8 \, {\rm{m/s^2}})}}{{-9 \, {\rm{m/s}}}}\)
Now, we can calculate the value of b:
\(b = -4.313 \times 10^{-5} \, \rm{kg \cdot m^{-1}}\)