A water droplet 4.0 mm in diameter is falling with a speed...

A water droplet 4.0 mm in diameter is falling with a speed of 10 km/h at an altitude of 20 km. Another droplet 6.0 mm in diameter is falling at 25% of that speed and at 25% of that altitude. The density of air at 20 km is 0.20 kg/m3 and that at 5.0 km is 0.70 kg/m3 . Assume that the drag coefficient C is the same for the two drops. Find the ratio of the drag force on the higher drop to that on the lower drop.

A water droplet 4.0 mm in diameter is falling with a speed of
10 km/h at an altitude of 20 km. Another droplet 6.0 mm in diameter is falling at 25% of that speed and at 25% of that altitude. The
density of air at 20 km is 0.20 kg/m3 and that at 5.0 km is 0.70 kg/m3
.
Assume that the drag coefficient C is the same for the two drops.
Find the ratio of the drag force on the higher drop to that on the
lower drop.

Key Concepts

Variation in Fluid Density with Altitude

Air density is not constant and varies with altitude. At higher altitudes, the density is lower, which affects the magnitude of the drag force according to the drag equation. Understanding how ?, the fluid density, changes with altitude is essential in problems where objects move through layers of the atmosphere with different densities.

Velocity Squared Dependence

The drag force depends on the square of the velocity. This means that even modest changes in the speed of a falling object can lead to large changes in the drag force acting on it. This quadratic relationship is fundamental in analyzing fluid dynamics problems involving moving bodies, such as water droplets falling through air.

Drag Force Equation

The drag force experienced by an object moving through a fluid is typically given by the drag equation, F_d = 0.5 * C_d * ? * A * v^2, where C_d is the drag coefficient, ? is the fluid density, A is the object's cross?sectional area, and v is the velocity relative to the fluid. This concept explains how these factors contribute multiplicatively to the resistive force that opposes the motion of the object.

Cross-sectional Area of Spherical Objects

For spherical objects like water droplets, the cross-sectional area (which is the area presented to the flow) is proportional to the square of the diameter. Specifically, the area is given by A = ?d^2/4. This relationship is key when comparing droplets of different sizes, as even small changes in diameter result in significant changes in the area and thus the drag force.

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