Speed Math

Two math puzzles:

#1: You have to drive to a meet someone at a specific time. Let’s say that you only have to drive on one road to get there, and the speed limit is the same the whole way. You calculate the time it will take to get there on time, and you start driving – but there’s traffic. So, you wind up driving half the distance at half the speed, then the traffic disappears.

How fast do you have to go the rest of the way to arrive at the meeting on time?

#2: You’re driving on a two-lane highway where the speed limit is 70 km/h. You are driving 100 km/h, and you pull into the left lane to pass someone who is driving the speed limit. Everything about the car you’re passing is identical to yours – even the driver weighs the same as you do. At the instant that you are side-by side, a train appears across the road in front of you and stops. You both hit the brakes at exactly the same time to try and stop from hitting the train.

Luckily, the person in the other car stops just as his bumper touches the train, let’s say 1 mm before touching it… But, because you were driving faster, you cannot stop in time.

How fast are you going when you hit the train?

The answers

#1. Most people instinctively say “double the speed limit” to make up the lost time. However, this is not the right answer.

Let’s say that the meeting is 100 km away, and the speed limit is 100 km/h. Therefore, it should take you 1 hour to get to the meeting.

If you drive half the distance (50 km) at half the speed (50 km/h), then at the moment the traffic clears up, you should have been at your destination. So, you would have to drive infinity km/h to get there. However, since teleportation doesn’t exist yet, you might as well just call and tell them you’ll be late.

#2: This one is a little tougher, but it should be pretty intuitive for someone working in audio. A car’s brakes work by taking the energy in the car’s momentum, and converting that to heat in the brake discs. The key word there is energy.

So, the question is: if you consider the amount of energy removed from the car going 70 km/h, and take that out of the energy in the car going 100 km/h, how much energy is left?

The answer is 70 km/h. For someone in audio, this might look like a familiar answer, since 0.7 V has half of the power of 1.0 V (assuming identical loads). In the case of the cars, it’s because the amount of power (the amount of energy that’s transmitted over time – in this case, to heat the brakes) to bring the car from 70 km/h to 0 km/h is identical to the amount of power it takes to bring the same car from 100 km/h to 70 km/h. (An audio geek might joke that 70 km/h is 3 dB slower than 100 km/h.)

The conclusion

Slow down. You’re not going to make it to the meeting anyway, and driving a little bit faster means you’re going to hit the train much harder than you think.