Rainy Day Mathematics
Dear Tom and Ray:
I've noticed something while driving, and I want to find out if it's real or just my imagination. While sitting at a stoplight during a light rain, a given amount of drops seem to hit my windshield. I turn the wipers on and off to clear them, say, every five seconds. When the light changes and I pull away, it seems as if more drops are hitting the windshield, and I need to either turn the wipers on and off more frequently or turn them on and leave them on. My questions: Are more raindrops hitting the windshield while moving? In effect, is the windshield larger as it travels through not only space, but time, making it gather more drops? Does the angle of the windshield have any effect? Am I imagining this in the first place? I have considered pasting construction paper to ping-pong paddles and running through a light drizzle, then counting the drops. Writing to you seems easier. I just need to get an answer. It's been bugging me for years.
TOM: We all carry burdens in life, Tim. Yours is a particularly heavy one. We are happy to help.
RAY: This is a very straightforward calculus problem, Tim, one that we are completely incapable of solving.
TOM: But we do know that the answer is yes, your windshield does get hit with more raindrops per second if you're moving forward.
RAY: The simplest explanation is that when you're moving, you are encountering not only the raindrops falling from right above you, but you're also intercepting some of the raindrops that would have fallen in front of you -- on your hood or in front of your car.
TOM: Since you brought up ping-pong paddles, let's work with that.
RAY: Imagine you hung a bunch of ping-pong balls from the ceiling in your garage to simulate a moment of a rain storm that's been frozen in time. So you have an even distribution of ping-pong balls -- maybe they're spaced one foot apart in every dimension.
TOM: Now, if you park your car in the garage, a certain number of those ping-pong balls will be in contact with your windshield, right? Let's just say there are five of them.
RAY: If this were a rainstorm, and those ping-pong ball drops were falling at the rate of one per second, you'd add another five drops to your windshield every second -- drops that are falling straight down onto your windshield from the sky. You with me so far?
TOM: But when you start to move forward, you're not only getting those five drops every second that are coming from right above you, but because of your forward motion, your windshield also is colliding with additional drops that would have fallen in front of your car had you been sitting still.
RAY: You're sort of adding a third dimension. You have the height and width of your windshield to accept whatever raindrops fall. But you're adding "depth" by plowing through additional regions of raindrops.
TOM: It might be easier to imagine if you think of an airplane instead of a car. Sitting on the tarmac, the plane would collect a certain number of raindrops on its windshield -- just like your car would. But if it were flying through that rainstorm at 550 mph, it would collide with a lot more of those drops, right?
RAY: Or to take it even a step further, imagine that the plane's windshield were missing completely. In which scenario do you think you'd get wetter? Sitting in that cockpit parked on the tarmac while it rained, or sitting there as the plane flew through the storm?
TOM: I think if I get on a plane and it has no windshield, I'm getting off and taking the train.