# A variant on a classic

One of @sparksmaths’s favourite puzzles is:

Two cars are driving along the motorway, one at 70mph and the other at 100mph. As they’re beside each other, they both spot an obstruction ahead and slam on the brakes. The slower car just barely stops before hitting the obstruction. Assuming no reaction time, equally efficient brakes and the general simplicity that goes with a mechanics problem, how fast does the other car hit the obstruction?

Someone on reddit – I’ve no idea who, and it’s too late in the day to check – suggested a variant: what if you take thinking time into account?

Spoilers below the line.

Let’s solve the original problem first. Assuming constant deceleration, we can use $v^2 = u^2 + 2as$ (in miles and hours, if we’re going to roll that way); $v=0$, $u=70$, and we really only want the $2as$ – which will be the same for the other car, and is related to the kinetic energy lost ((assuming unit mass)).

So $2as$ bit is just -4900 miles squared per hour.

The other car is travelling at 100mph, so its unit-mass energy is 10,000 units before it starts braking, and 5,100 units afterwards. If you’ve not done the puzzle before, it’s surprising that the second car hits the obstacle at about 71.4mph ((Interestingly, $\sqrt{5100}$ is very close to $70+\sqrt{2}$. Maybe the Ninja will explain why one day.))

Now for the trickier version. While we probably *could* work it all out, as I mentioned, it’s a bit late in the day, so I’m just going to look up the Highway Code numbers, which say that at 70mph, it takes 21m before you start to brake, and then 75m to stop. I’m as annoyed about the change of units as you are, ok?

In any case, that means the faster car travels 30m before it starts to brake – so it’s only 66m from the obstacle at that point. It has 12% less distance to stop in than the slower car, so it loses 12% less energy while braking.

So, rather than losing 4900 units, it loses a smidge over 4300 and ends up with 5688 units. To my surprise, that’s only a little faster than before, about 75.4mph.

That said, hitting things at 75.4mph is the sort of thing only the Mathematical Ninja would consider doing, and even then only things they *really* disliked. Say, what’s happened to my calculator?