u/johsua_banggg

Hello friends, I was recently here asking about the paradox of Achilles and the tortoise. I got some helpful (and some unhelpful) responses, but I still have some questions, so I wanted to revisit the question with a more concrete example.

To make the numbers extremely simple, let's suppose that Achilles runs at a constant velocity of vA = 2 meters per second and the tortoise runs at a constant velocity of vT = 1 meter per second. Let's suppose that they instantaneously accelerate from rest to top speed with perfect reaction time. Let's suppose that Achilles starts the race at x1 = 0 meters, the tortoise starts at x2 = 10 meters, and the finish line is xf = 20 meters.

Analyzing from the intuitive, physics-based perspective, the time that it takes for Achilles to finish the race is:
> tA = (xf - x1)/vA
> = (20 - 0)m/2m/s
> = 10 seconds

The time it takes for the tortoise to finish the race is:
> tT = (xf - x2)/vT
> = (20 - 10)m/1m/s
> = 10 seconds

So according to this intuitive analysis, Achilles and the tortoise tie the race at 10 seconds.

But now consider this. At t0 = 0s, Achilles is at xA0 = 0m and the tortoise is at xT0 = 10m. Achilles must catch up to the tortoise: xA1 = 10m. At this point, the time is:
> t1 = t0 + (xA1 - xA0)/vA
> = 0s + (10 - 0)m/2m/s
> = 5s

During this time, the turtle has travelled to:
> xT1 = xT0 + vT\*(t1 - t0)
> = 10m + 1m/s\*(5 - 0)s
> = 15m

So at t1, Achilles has not yet caught up to the tortoise. Doing the process once more:
> xA2 = xT1
> = 15m
> t2 = t1 + (xA2 - xA1)/vA
> = 5s + (15 - 10)m/2m/s
> = 7.5s
> xT2 = xT1 + vT\*(t2 - t1)
> = 15m + 1m/s\*(7.5 - 5)s
> = 17.5m

At t2, Achilles has not yet caught up to the tortoise. Once more:
> xA3 = xT2
> = 17.5m
> t3 = t2 + (xA3 - xA2)/vA
> = 7.5s + (17.5 - 15)m/2m/s = 8.75s
> xT3 = xT2 + vT\*(t3 - t2)
> = 17.5m + 1m/s\*(8.75s - 7.5)s
> = 18.75m

At t3, Achilles has not yet caught up to the tortoise.

Can someone please clarify this? Looking at the pattern, it seems that Achilles will never catch up to the tortoise, even with more rigorous analysis.

Hello friends, I have a question regarding a certain paradox, and I was wondering if someone could help me solve it.

Achilles and a tortoise enter a race. Since Achilles is faster than the tortoise, the tortoise gets a headstart.

When the race begins, Achilles must first reach where the tortoise was previously. But by the time he gets there, the tortoise has moved forward. So Achilles must then reach the tortoise's second position, at which point the tortoise has moved even further forward.

Intuitively, you would think that Achilles will win the race since he is faster than the tortoise. However, the analysis seems to show that Achilles can never catch up to the tortoise, let alone beat him in the race.

Can anyone help me solve this? Who wins the race?