Question

Calculate the velocity a spherical raindrop would achieve falling (take downward as positive) from 4.2 km in the following situations.

(a) Calculate the velocity in the absence of air drag.

(b) Calculate the velocity with air drag. Take the size across of the drop to be 4.8 mm, the density of air to be 1.17 kg/m3, the density of water to be 1000 kg/m3, the surface area to be πr2, and the drag coefficient to be 1.0.

Answer 1

To calculate the velocity of a raindrop falling from 4.2 km without air drag, use the formula v = sqrt(2gh). With air drag, the terminal velocity occurs when the gravitational force equals the drag force, for which the drag equation and gravitational force need to be equated and solved for v.

Step-by-step explanation:

To calculate the velocity of a spherical raindrop falling from 4.2 km, we consider two cases: without air drag and with air drag.

Without Air Drag

In the absence of air drag, the velocity can be found using the kinematic equation v = sqrt(2gh), where g is the acceleration due to gravity (9.81 m/s²) and h is the height (4.2 km or 4200 m). Thus, v = sqrt(2 * 9.81 * 4200) gives us the velocity.

With Air Drag

With air drag, the terminal velocity is when the force of gravity is balanced by the drag force. The drag force can be calculated using the drag equation, Fd = 0.5 * Cd * rho * A * v², where Cd is the drag coefficient, rho is the density of air, A is the cross-sectional area (calculated using pi*r²), and v is the terminal velocity. Given the size of the raindrop (4.8 mm or 0.0048 m in radius), the density of air (1.17 kg/m³), the density of water (1000 kg/m³), and a drag coefficient of 1.0, we can solve for v. The cross-sectional area is pi*r². The equation to find terminal velocity equates the gravitational force (mg) to the drag force (Fd). By rearranging and solving for v, we find the terminal velocity.