Accidents and estimates

I came very close to being in a car accident on Friday, which got me thinking about kinematics and estimation in engineering calculations.

I was stopped in a line of cars at a traffic light when something—probably the squeal of brakes, although I may have heard that after later—made me look up into my rear view mirror. A car going way too fast was coming up behind me. I leaned back in my seat, put my hands on my head, and closed my eyes, waiting for the impact.

Which never came.

After a couple of seconds, I opened my eyes and looked in the mirror. There was a car back there, slanted out toward the shoulder, but it was much further away than I expected. Then I noticed a car on the shoulder to my right and ahead of me. That was the car I had expected to hit me. The driver had managed to swerve to the right and avoid me.

That led to some conflicting feelings. I was pleased he was skillful enough to steer out of the accident but angry at his stupidity in needing to exercise that skill. Then the engineer in me took over. If he came to a stop ahead of me, how fast would he have hit me if he hadn’t veered to the right?

It’s a pretty simple calculation, the kind you learn in high school physics. There are two equations of kinematics we need:

[d = v_0 t - \frac{1}{2} \alpha g t^2]

and

[v_0 = \alpha g t]

These are covering the period of time from when his front bumper passed my rear bumper to when he came to rest. The distance traveled is [d], his speed at the beginning of this period is [v_0], the duration is [t], and the deceleration (assumed constant) is [\alpha g]. It’s common in situations like this to express the acceleration or deceleration as a fraction of the acceleration due to gravity; [\alpha] is a pure number.

We don’t really care about [t], so with a little algebra we can turn these into a single formula with only the variables of interest:

[v_0 = \sqrt{2 \alpha g d}]

Based on where the car ended up, I’d say [d] is about 25 feet. The deceleration factor, [\alpha] is a bit more dicey to estimate, but it’s likely to be somewhere around 0.6 to 0.8. And since we’re using feet for distance, we’ll use 32.2 [\mathrm{ft/s^2}] for [g]. That gives us a range of values from 31 to 36 [\mathrm{ft/s}] for [v_0]. Converting to more conventional units for car speeds, that puts him between 21 and 24 mph. That would have been a pretty good smack. Not only would my trunk have been smashed in, I likely would have had damage to my front bumper from being pushed into the car ahead of me.

This was a simple calculation, but it illustrates an interesting feature of estimation. Despite starting with a fairly wide range in our estimate of [\alpha] (0.8 is 33% higher than 0.6), we ended up with a much narrower range in the estimate of [v_0] (24 is only about 15% higher than 21). For this we have the square root to thank. It cuts the relative error in half.

Why? Let’s say we have this simple relationship:

[a = \sqrt{b}]

We can express our uncertainty in the value of [b] by saying our estimate of it is [b (1 + \epsilon)], where [\epsilon] is the relative error in the estimate. We can then say

[\sqrt{b(1 + \epsilon)} = \sqrt{b}\sqrt{1 + \epsilon}]

and using the Taylor series expansion of the second term about [\epsilon = 0], we get

[\sqrt{b} \left( 1 + \frac{1}{2}\epsilon + \mathrm{h.o.t} \right)]

If the absolute value of [\epsilon] is small, the higher order terms (h.o.t) won’t amount to much, and the relative error of [a] will be about half that of [b].

Lots of engineering estimates aren’t as forgiving as this one, so it’s important to know when your inputs have to be known precisely and when you can be a little loose with them.

Speaking of forgiving, I searched for rear end crash test results for my car to see how much damage it would have taken. I came up empty, but here’s a more recent model undergoing an impact at twice the speed.