Run or Walk under the rain?
Looking through my college notes to find something interesting I found that problem that I solved for fun while assisting to a boring class. I know it is kind of geek, but anyway…
So in order to solve that problem we need first to make some assumptions.
1-The rain falls at constant speed and density. 2-The rain falls vertically and at constant angle. 3-There are no splashes. 4-You are living in an 8bits world where you only have 2 dimensions and you are essentially a rectangle.
So you want to start from point A to point B by minimizing the rain received. To do that we need to consider two vectors:
1-The velocity of the rain. (The vertical ones.)
2-Your velocity. (The horizontal ones.)
We will simply move the vectors so that we see where the water drops affecting you are. There are some hitting you in the front and some falling on your head. For one displacement, the red area represent the amount of water that fall on your head, while the blue one represent the amount you will receive from the front. In this example, it takes about 5.5 displacements, so 5.5 times those area will be the total amount of water received.
Let’s suppose you go faster this time, there is what happens:
In this example it would take about 1.25 displacements to travel from A to B. Obviously if you go from A to B at infinite speed, the area of water touching you on the top will become null. So at least now you know that there is a way to preserve your hairs under the rain! But that doesn’t take into account the area of water in front of you, so let’s do the maths!
The area of a parallelogram is given by: Area = Base * Height
So the Areas of our parallelograms are:
Top_Area = Your_Top * Rain_Velocity
Front_Area = Your_Front * Your_Velocity
The total amount of water you will get depends on the sum of the two areas and the number of cycle to reach destination.
Total_Water = Nbr_Cycles * (Top_Area + Front_Area)
The number of cycles is equal to:
Nbr_Cycles = Distance_To_Travel / Your_Velocity
So we can rewrite the Total_Water equation as:
Total_Water = Distance_To_Travel / Your_Velocity* (Your_Top * Rain_Velocity + Your_Front * Your_Velocity)
Using distribution and after performing simplification: (Here I assumed that Your_Velocity>0 which is obvious as otherwise you would never reach your destination and get all the rain on your head!)
Total_Water = Distance_To_Travel * Your_Top * Rain_Velocity / Your_Velocity + Distance_To_Travel * Your_Front
The only variable in this equation is Your_Velocity that we can optimize to reduce Total_Water as the other parameters are all constant by the initial assumptions.
So water you get on the front is constant and is: Distance_To_Travel * Your_Front. So there is nothing we can do about it. However we can minimize the water falling on your head by increasing Your_Velocity toward the infinity.
Conclusion it is favourable to run as fast as possible under the rain but even thus you can relatively protect your hair, you’ll always get some water in the face! Next I redo the exercise but I will add an inclinable umbrella in the equation! (But by pure logical deductions, you would run at infinite speed and place your umbrella vertically to protect you from the Front_Area!)
