We are asked to determine the velocity of a rain drop if it falls from 4 km.
To do that we will use the following formula:
Where:
If we assume the initial velocity to be 0 we get:
The acceleration is the acceleration due to gravity:
Now, we take the square root to both sides:
Now, we substitute the values:
solving the operations:
Therefore, the velocity without air drag is 280 m/s.
Part B. we are asked to determine the velocity if there is air drag. To do that we will use the following formula:
Where:
We need to determine the drag force. To do that we will use the following free-body diagram:
Since the velocity that the raindrop reaches is the terminal velocity and its a constant velocity this means that the acceleration is zero and therefore the forces are balanced:
Now, we determine the mass of the raindrop using the following formula:
Where:
The volume is the volume of a sphere, therefore:
Since the diameter of the raindrop is 3 millimeters, the radius is 1.5 mm or 0.0015 meters. Substituting we get:
Solving the operations:
Now, we substitute the values in the formula for the drag force:
Solving the operations:
Now, we substitute in the formula:
Now, we solve for the velocity:
Now, we substitute the values. We will use the area of a circle:
Substituting the radius:
Solving the operations:
Now, we take the square root to both sides:
Therefore, the velocity is 8.4 m/s
