Gasoline Paradox: Car Can't Run out of Gas?

Question

I have heard of a statement like this:

A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time.

Is this possible (mathematics wise)?

Answer 1

Assume that the car engine is perfect and proportionally converts gas into distance. Then the answer is yes and no. Using the series $$ \sum_{j=1}^\infty \frac{1}{2^j} = 1 $$ The car never runs out of gas since infinitely many terms are non-zero, but only travels a finite distance since the sum is finite.

This only works if you view gasoline as continuous matter instead of discrete particles.

Answer 2

You are overthinking this. Yes, in draining a 100 litre tank full of gasoline, you can imagine that infinitely many events occur: At some point in time for example, the tank will be (1) 1/2 full, and (2) 1/4 full, and (3) 1/8 full, and so on. But we can measure the rate at which the tank is being emptied, in units of say litres per hour, and calculate precisely when the tank will be emptied even if infinitely many of the above "events" had occurred in the interval. Nothing "paradoxical" here.

Answer 3

"A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time."

A more practical restatement: If you reduce the speed of the car as a function of time like $v(t) = v(0)\exp(-a t)$, then the car will always keep on moving, yet the fuel consumption will be bounded.