# Surprising result of calculating speeds of two cars hitting a tree

Originally published at: https://boingboing.net/2018/01/22/surprising-result-of-calculati.html

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someone post the “surprising result” I aint got time for watching that video… but I must know!

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The answer is 100 MPH. The red car driver’s mistress, driving the blue car, cut his brake line because he refused to leave his wife.

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I do say that people that tailgate on the freeway has no understanding or respect for physics. Over the years I got to understand just how hard it is to stop at high speeds and these days I keep very respectable stopping distances.

Inertia is a bitch.

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Spoilered TL;DW and explanation: Assuming equal mass, the 100 mph car has roughly twice as much kinetic energy (v^2 ) as the 70 mph car. Work done slowing the car is F*D. D is the same for both, and if the brakes are as good, so is F. So the 100 mph car loses roughly half it’s energy – putting it at 70 mph.

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It used to trouble me when I heard people say, “It’s important for everyone to learn physics because people need to know the relation between kinetic energy and velocity when they’re driving!” Because really, you’re not going to run calculations through your head when you’re driving along and all you really need to know is that the faster you’re going, the longer you’ll need to stop.

I say “used to” because I haven’t actually heard anyone talk about the importance of learning physics for a while now.

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More like they need to get a better understanding of the energy involved. They need to learn to respect how they are probably not going to easily stop a car at speed without some distance. What that does to your body trying to get rid of all that energy.

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Keep in mind that by the time the blue car stops, the red car has already hit the tree.

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That is not a good first approximation.

First off, the red car is going to impact the tree before the blue car comes to a halt, so the brakes aren’t going to have as much time to work for the red car as the blue car.

Far more importantly, braking deceleration is limited by how quickly the car can decelerate without going into a skid. That’s guesstimated here as about six meters per second per second. With constant deceleration, proportionally more kinetic energy is bled off (converted to heat) in the moments just after the brakes are applied-- potentially giving hope that even though the red car has less time to brake, the time it DOES have might serve convert to heat the lion’s share of the kinetic energy.

Hence, there is no substitute for doing the math.

The blue car going 70 miles per hour is doing 31.3 meters per second. Decelerating at 6 m/s, that’s going to take 5.22 seconds to come to a complete halt, covering a distance of (.5 a t^2) = (.5 * 6 * 5.22^2) = 81.6 meters. (It’s the same amount of time as it would take to accelerate from a standing start to 31.3 m/s at an acceleration of 6 m/s^2.)

The red car going 100 miles per hour is doing 44.7 meters per second. The car’s speed at time t is v = 44.7 - 6 * t. Integrate that with respect to t and set the position at time 0 to zero, we get d = 44.7 * t - .5 * 6 * t^2. We know d is 81.6 meters, so solve for t: 3 t^2 + 44.7 t - 81.6 = 0. Plug that into the quadratic equation, the only positive root is t = 1.64. So the red car’s velocity at impact is 44.7 - 6 * 1.64 = 34.9 meters per second, or 77 miles per hour.

EDIT: except I biffed a sign, there. The relevant equation is -3 t^2 + 44.7 t - 81.6 = 0. Solve for that one and you get t = 2.13 or 12.8 (with the latter corresponding to passing the tree, slowing to a stop, then accelerating back towards the tree, at at a constant rate of -6 m/2^2). Speed at t = 2.13 is 31.9 m/s, or 71 miles per hour-- exactly what the fellow in the video approximated.

Surprised, I ran the numbers again for a slippery road with maximum deceleration of -3 m/s. 162 meters and 10.4 seconds for the blue car to brake, corresponding equation for the red car is -1.5 t^2 + 44.7 t - 162 = 0, roots 4.22 and 25.6 seconds. Speed of the red car at t=4.22 is 32.0 m/s, which is basically 71 miles per hour again, given the rounding errors.

I guess there’s an argument to be made that this is a consequence of energy = force * distance, where we know the force is the same for whatever the mass of the car and slipperiness of the road allow…

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Measuring your distance behind with the car in front as you both pass the same point with the phrase “Only a fool breaks the two second rule” is a good guide.
Distance IS Time.

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Actually, drivers’ manuals often now say THREE seconds, for the “following rule”, a big change from when I was a kid o.O’. Not that many people follow that advice, of course…

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Braking time is only one of many very good reasons to not tailgate.
Here are some others:

• greater visibility - you can see more of the road when you aren’t using the car in front of you to blot it out
• energy efficiency - braking is the definition of wasted energy, giving yourself the space to decelerate is more efficient
• traffic - crowding other drivers is the primary cause of traffic creation, according to these mathmaticians: https://phys.org/news/2017-11-buffer-bumper-contradicts-traffic-tailgating.html
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We’re too busy trying to prevent people from eating laundry detergent.

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That sounds like an upgrade.
Yeah, in my experience most people could just about say “Only…”

Guess I’m not the only one who presses the imaginary brake pedal as a passenger, huh?

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See, this is why I wished I’d paid more attention in physics at school. Once I’d seen the video far enough to learn the answer (and be surprised at it), I managed to derive it myself with a whole mass of horrible equations that finally, after cancelling out the unknowns (distance to tree and braking deceleration), expressed the final velocity of the 100 mph car in terms of the other car’s starting and final velocities. It would never have occurred to me to think about it the way you did.

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We don’t HAVE to stop them. It’ll all come out in the wash.

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glare

We have ways of dealing with…YOUR kind.

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In Ontario, it’s 2 seconds on the multiple-choice part of the exam, and 3 seconds will be marked wrong. Give them the answer they want but drive more cautiously.

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Good point; I should have said “US state-supplied driving manuals”. My apologies to Canuckistan ^^.

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Then I wash my hands of the whole situation.

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