homework and exercises - What is the force required to move a raindrop?

I'm doing some research on how much force is required to move a typical raindrop (assuming that it is falling straight down) off to the side of X distance. This is for a school project on creating our own airbrella.

Doing some reasearch it seems a typical raindrop is 4-100MG and falls at about 5-20MPH. Also I would like to know how far the raindrop would be blown for X amount of wind. Assuming the wind is hitting it at a 120 degree angle? (im guesstimating 120 because of the picture below)

Answer

What is the force required to move a raindrop?

The force $F_{(t)}$ that pushes against the drop at time $t$ is

$$F_{(t)} = \frac{cW\cdot A\cdot \rho\cdot v_{(t)}^2}{2}$$

where $cW$ is the drag coefficient, $A$ the cross section area of the drop, $\rho$ the density of the air and $v_{(t)}$ the velocity at any given time, but what you might be asking for is the initial kinetic energy, which is $M\cdot v_{0}^2/2$ where $v_0$ is the initial velocity of the drop.

To calculate how far the drop moves with the kinetic energy you give it initially, I would solve this problem with the ballistic equation, which requires a differential equation if you want to have air resistance:

Code (Mathematica, .nb File)

Here the mass of the drop is 0.004 gram, its initial velocity is 10 m/sec, the initial angle 10° and the height from which it is thrown 1 meter. It is assumed that the drop has 1 mm radius and a cW value of 0.3:

Then the drop would fly 0.67 sec and reach the ground after 4.7 meter.

I might put this into Latex when I have the time later, but in the end you will need a math program anyway since differential equations are hard to solve by hand and in this case there is no analytical solution as far as I know.

If you choose unrealistically high initial velocities you also have to note that the waterdrops would vaporize, and if they are to big they will split up. This is not taken into account in my calculation. More information on drops, their mass and shape (which influences the cW value) can be found here and here.

Edit: put in more realistic values for the drops