"How long 'til we get there?" Road trip puzzle

Road trips can be fun, but they often appear to go slower the closer you get to your destination. I thought up this puzzle while on a recent trip. Thought it would be good food for thought. Curious about the different approaches to solving it.

Suppose you have D miles until you reach your destination. The rule is that the speed at which you travel is equal to the distance to your destination. So when you are 60 miles from your destination your speed must be 60 mph; 50 miles from destination, 50 mph; etc.

How long until you reach your destination?

EDIT: I'm pretty sure that the answer is infinity—you will never get to your destination because you will always be one hour away. I'm curious about how people come up with their solutions. So far, very entertaining.

Solution 1:

Your speed is strictly decreasing, so at any time, you know that it takes you at least one hour which would be the arrival time at present speed. So, you cannot reach the destination which is always more than an hour away.

Solution 2:

Put a car following the same rules but which is exactly in the middle between you and your destination, at $D/2$ miles from the destination. It is moving at half of your speed so it stays exactly at half the distance between you and your destination, at all times. So, supposing your journey ends, the other car will arrive exactly at the same time as you at the destination.

However you quickly realize that, since you follow the same rules, the total duration of your trip is the time it takes you to cover half the distance to your destination + the total duration of his trip. So you will arrive strictly after him.

So supposing that the journey ends gives a contradiction : you will stay on the road forever.

Solution 3:

The existing answers are very good and don't require calculus; nevertheless it seems worthwhile to also show how this would be solved using calculus.

If your position at time $t$ is $x(t)$, your velocity (in appropriate units) is $\dot x(t)=-x(t)$. The general solution of this linear first-order ordinary differential equation is $x(t)=c\,\mathrm e^{-t}$, with arbitrary constant $c$. Thus you get arbitrarily close to your destination, but your speed decreases exponentially and you never reach the destination.

Solution 4:

The problem can be written as linear ODE of order 1:

Lets say that $s(t)$ is the driven distance, and $v(t)$ is the speed. Then we have $v(t)=s'(t)$. On the other hand, we have the relation $v(t)=60-s(t)$ and therefore $$s'(t)=60-s(t).$$ This is a linear ODE of first order with constant coefficients, and its solution is $$s(t)=60-60e^{-t}.$$ Now you want $s(t)=60$, so we have $$60=60-60e^{-t}$$ and therefore $$e^{-t}=0.$$ There is no value $t\in\mathbb R$ that fulfills this equation, only the limit $\lim_{t\rightarrow\infty}e^{-t}=0$.