Question

QUESTION 3 [10]

With this problem, we want to explore the idea that, if it were not for drag, raindrops would attain fantastic speeds near Earth’s surface. Piet observes that the raindrops that are hitting him, have a radius of 2.00 mm and fall from a cloud located 1000 m above the ground he is laying on. Take the drag coefficient of the raindrops to be 0.50 and the ambient temperature to be 20.0 °C.

HINT: The area in the terminal velocity formula is a cross-sectional area of the object under consideration.

(a) Determine the terminal speed of the raindrops. You may assume spherical raindrops made out of water only. (4)

(b) Piet notices that the expression for the terminal speed used above does not contain explicit reference to the height of the cloud from whence the raindrops originate. Would the terminal speed for the raindrop in (a) be equal to that for a raindrop released 1.00 m above ground? Is there a discrepancy? Give reasons? (3)

(c) Determine the final velocity of the raindrop (in km/h) in when we ignore air resistance (3)

Answer 1

Solution of (a):

We have,
Radius of the raindrop
Height of the cloud
Drag coefficient of the raindrop
Density of the raindrop

The terminal velocity is calculated as:

Where is the area of the raindrop, is the weight of the raindrop, is the mass of the raindrop, and is the gravitational acceleration, and is the density of air.

Mass of the raindrop is given in terms of its volume and density :

Therefore, the weight of the raindrop is

Substituting the value of equation (2) into equation (1):

Therefore, the terminal speed of the raindrop is .

Solution of (b):

The speed of raindrop released from the height is calculated from the kinematics formula:

Since raindrop is initially at rest, the initial velocity .

There is large discrepancy between the raindrop speed released from the height 1000 m above the ground and raindrop speed released from the height from the 1 m above the ground. This discrepancy arises due to: in the calculation of terminal speed of raindrop which released from 1000 m, the height is not considered and resistance of air is considered while in the calculation of final speed of the raindrop which released from 1 m, the height is considered and resistance of air is not considered.

Solution of (c):

The final speed of raindrop released from the height is calculated from the kinematics formula:

Since raindrop is initially at rest, the initial velocity .