How fast do you need to drive in the rain to keep your rear window dry?

November 09, 2011 By: erik Category: Geeky, Math 1,579 views

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Car Sketch - Rear Window AngleLast week, we had quite the deluge, which coincided, unfortunately, with me having to drive 120 km. I noticed that whenever I slowed down, the rear window would get wet and hard to see out of. Once I sped up again, one wiper pass dried it off and it stayed dry until I slowed down again. This is an obvious scenario for anyone who has ever driven, or anyone who thinks briefly about the physics involved: when you’re driving fast, the rain doesn’t hit your rear window. Out of the blue (or gray, in this case), my mouth spoke the words, “I wonder what’s the minimum speed I have to drive to keep the back window dry?” My wife immediately intuited that “it depends on the shape of the car”, by which she meant the angle of the rear window. This insight made the original question all the more interesting to me, because it gave me an equation to plot! Are you getting excited yet?

Let’s define the angle of the rear window as being the angle, going down, from horizontal, like so:

Car Sketch - Rear Window Angle

I did not draw this car, René van Belzen did.

If we stop to think about it – and we should, as it’s good practice – two features of the plot of our “minimum speed to keep the rear window dry given the angle θ” equation can be deduced. Firstly, when θ is 90°, the minimum speed will be zero, since the rain – which we are assuming, for simplicity, is falling straight down – won’t hit it even if we are stopped. Secondly, as θ approaches 0°, the minimum speed to keep the rear window dry will shoot off to infinity. Note that we shall also be ignoring near-speed-of-light Einsteinian physics, as the Newtonian model is more than sufficient for the speeds I drive. So we can picture our graph dropping down from infinity and then making its way down to the x-axis, maintaining a negative slope the entire time, until it hits 90° on the x-axis.

The time it takes the rain to fall the vertical height of our window is going to be the height divided by the terminal velocity of the rain. And the time it takes the car to move the horizontal length of our window is going to be the length divided by the velocity of the car. When those two times are equal, the car is traveling at the minimum velocity to keep the window dry.

Rain - Car Equation

Okay, so we have our equation, but we have too many variables. We need to find a way to write h in terms of l, and we need a rain velocity.

Simple trigonometry tells us that the tangent of the angle in a right triangle is equal to the opposite side divided by the adjacent side, so…

Tangent of Theta

Thus we can replace h in our previous equation, giving us:

Car Rain Equation

The length cancels out, and we get our final equation:

Car Rain Equation

There are two things that I like about this. Neither the height nor length of the window really matter, just as my wife’s brain intuited instantly. The second is that of course the tangent operation is involved, since it zips off to infinity, yet calms down near the x-axis. Since we need it to zip off to positive infinity near the y-axis, it must be inverted. What I love about math and physics is just how intuitive it can become. My mathematician readers were probably already thinking about the inverse tangent function when I discussed the general shape of the data, particularly given that an angle – and thus trigonometry – was involved. Let’s continue!

Wait, before we continue, let me go off on a bit of a tangent (oh dear, trigonometry jokes are a bad sine)… After I finished writing this post and was lying in bed trying not to think about angles and hypotenuses, it occurred to me that the tangent only has to be inverted because of my arbitrary definition of where we start measuring the angle. If we define 0° as being the vector pointing straight down, the direction the rain is falling, and measure the angle away from the movement of the car, then we get a plot that starts at 0 mph at 0° and goes up to ∞ mph at 90°, just like the normal, unmodified tangent function does! But wait, this mathematical model gets even better! What happens if you increase the angle past 90° (sort of like a spoiler)? The tangent function flips back to negative infinity and comes up to 0 mph at 180°, which still accurately models the speed the car would have to travel to keep the rear window dry, since it would have to be going very fast in reverse, and then less fast as the angle approaches the vertical 180°! This is exactly the kind of real world application of a mathematical principle that is so often lacking in our education system. Anyway, I thought that was cool. Back to our regularly scheduled post…

We need a velocity for the rain. What is the terminal velocity for a rain drop? I asked the internet, and the internet said that a small drop of light stratisform rain falls at about 4.6 mph, and the largest possible drop – they can’t be bigger than 5 mm or they break up (awesome high speed video!) – from a huge thunderhead falls at up to 20 mph. We’re going for a minimum speed to guarantee a dry rear window, so we’ll use the 20 mph figure.

When we substitute 20 mph for our rain velocity and convert from radians to degrees, our final equation looks like this:

Rain Car Equation

And now we’re ready to plot it! Drumroll please…

Minimum Driving Speed to Keep Rain Off Rear Window

Exactly what we expected, isn’t it?

Examples

57° Rear Window Angle

This Wolsely Hornet has a rear window angle of 57°. That means that, if it goes more than 13 mph in even the harshest thunderstorm (of absolutely vertical rain), no rain will hit its rear window. Isn’t that incredible? It seems slow to my internal physics engine.

35° Rear Window Angle

This Peugeot 203 Limousine has a 35° rear window, so it will need to accelerate up to 28.5 mph in order to keep it dry. As you would expect, if you have a 45° window, the velocity you need to travel is 20 mph, exactly equal to the rain’s velocity.

16° Rear Window Angle

And lastly, this Porsche has a 16° rear window, and thus it will need to zip up to 70 mph to keep dry. Obviously this will be no problem for the Porsche, but you can see how the velocities are increasing very rapidly as we approach the horizontal, due to the very nature of the tangent function.

My curiosity is satisfactorily quenched on this topic, and I hope you learned something. Math is all around us and lets us know things about the world through mere thought and some pen and paper. Isn’t that fascinating?

 
  • aquariumdrinker

    In my experience, a pickup truck’s vertical rear window will get wet when the pickup is driving through vertical rain.

    • http://erikras.com/?utm_source=disqus&utm_medium=profile&utm_campaign=Disqus%2BProfile Erik R.

      Surely that’s from roof runoff and not a direct cloud-to-glass hit.

      • Josh Grady

        See, that’s the old mathematician/engineer dichotomy raising its ugly head.  The engineer would want to take the effect of the car moving through the air into account.

        “…we shall also be ignoring near-speed-of-light Einsteinian physics, as the Newtonian model is more than sufficient for the speeds I drive.”  Needs to be reworked as a bumpersticker.  

        Great post!

        • http://erikras.com/?utm_source=disqus&utm_medium=profile&utm_campaign=Disqus%2BProfile Erik R.

          Now that I am imagining a high speed camera in a windtunnel to visualize this, I remembered that the Mythbusters did a similar thing testing whether you could drive an open convertible fast enough in the rain to keep the interior dry. They decided it was plausible.

      • aquariumdrinker

        Can we just delete the version of this where my paragraph breaks got deleted?

        Once the truck gets moving, it’s mostly from turbulence in the rain-bearing air through which the truck is moving. This is not really within the scope of your post, but it is a problem that even on a perfectly still day, the air passing over the back window of a moving car is not still and will become turbulent when speeds are great enough. Those speeds are very high for the Porsche. They are much lower for the pickup and they would probably be pretty low for the Wolsely Hornet if that were a real car and not something invented by MGM as a Munchkinland prop for the Wizard of Oz.

        My car has a plastic piece that overhangs the very top of the rear window. (I think of it as being there to provide shade. I hate to imagine that anyone thought the RAV4 would need a spoiler to keep the rear wheels firmly planted during high-speed maneuvering.) If I get above about 50 mph in the rain, there are sprayed droplets to be found under that overhang.

        I was thinking earlier it would be interesting to look at this as a way to determine the volume of rain per minute that would hit a square foot of back glass given the rain flow, the window slant and the car speed. I think your approach would let us rough out an answer, but I think the quality of our results would drop off quickly for high angle windows at high speeds.

        I’d also be curious to see whether we could use the delta between predicted and actual results to ferret out the impact of airflow features other than turbulence. For example, the Audi TT looks a lot like an airfoil in cross section. At high speeds, I bet the creation of expanding and quicker layers of air near the rear window would change the shape of the region in space that contains the drops that will eventually hit the window. I lack the formal training in Not Being Lazy to develop a sense for whether that space could get bigger or smaller as a result, or whether its volume is necessarily fixed.

        Awesome post. 

        • http://erikras.com/?utm_source=disqus&utm_medium=profile&utm_campaign=Disqus%2BProfile Erik R.

          First we need to pitch our show to the Discovery Channel as a combination of Top Gear, Mythbusters and Time Warp. Then it’s just a matter of setting up the wind tunnel and high speed cameras. The car manufacturers will probably pay us to use their vehicles in the tunnel.

          We need a name. Crazy Eddies? Tunnel Vision?

          • aquariumdrinker

            “…aaaand welcome back. We have a real treat for you this week. We’re going to do the exact same thing as every other week but with a slightly different car.” Sounds as good as most other basic cable programming.

          • http://erikras.com/?utm_source=disqus&utm_medium=profile&utm_campaign=Disqus%2BProfile Erik R.

            It’s all about snazzy graphics, a good narrating voice (did you know the Mythbusters narrator lives and does all his narration work from Australia?) and some attractive, charismatic nerds. Or, if we wanted to go the Top Gear route, some pompous, but knowledgable, assholes.

          • aquariumdrinker

            I’m hoping to bust the ugly pompous faux intellectual niche wide open.

  • Armen

    Thanks for this. This concept was a question on my physics exam, and you helped clear it up :]