# What is the limiting value of his average velocity?

Each second a rabbit moves half the remaining distance from his nose to a head of lettuce. Does he ever get to the lettuce? What is the limiting value of his average velocity?

I dont think he ever gets to the lettuce, just very close to it. Is the limiting value of his average velocity instantaneous velocity? So basically as time increases, both his velocity and position decrease?

Thanks

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#### jim mcnamara

Mentor
This thing is called Zeno's (also spelled Xeno's) paradox. In the real world, the rabbit actually does eventually eat the lettuce (you can't get much closer than that). If that is true, then there must be something about the way the "problem" is stated that leads you down to your conclusion that he never really touches the lettuce, just gets ever closer.

As the problem is stated, you are correct. The difficulty lies in accepting the premise that with each movement he gets one-half of the previous distance closer to the lettuce.

I would assume that his average velocity is 0, since it takes him an infinite time to reach the lettuce. As for instantaneous velocity, lower limit would be 0, upper limit I am not sure of, I think it would be highest on the first hop towards the lettuce.

Regards,

#### HallsofIvy

Homework Helper
Each second a rabbit moves half the remaining distance from his nose to a head of lettuce. Does he ever get to the lettuce? What is the limiting value of his average velocity?

I dont think he ever gets to the lettuce, just very close to it. Is the limiting value of his average velocity instantaneous velocity? So basically as time increases, both his velocity and position decrease?

Thanks
Actually, the way this was phrased, it is not Zeno's paradox! Here we are told the rabbit "moves half the remaining distance from his nose to a head of lettuce". It doesn't say the rabbit moves again!

#### NateTG

Homework Helper
HallsofIvy said:
Actually, the way this was phrased, it is not Zeno's paradox! Here we are told the rabbit "moves half the remaining distance from his nose to a head of lettuce". It doesn't say the rabbit moves again!
You are correct in that it is not Zeno's paradox, but the problem suggests that the rabit continues to move towards the lettuce - that's why it's "each" rather than "this" second.

The reason this is not Zeno's paradox is twofold - fundementally because the rabit is actually slowing down while, in Zeno's paradox, the rabit moves at a constant velocity, and, semantically, because we all know that Zeno's paradox is all about Achilees and a Tortise racing ;)

Regarding the original post:
The rabit does not get to the lettuce. Perhaps you could figure out how far from the lettuce the rabit is after 1 second, 2 seconds, 5 seconds, or x seconds? The rest of the post is a bit confused.

#### HallsofIvy

Homework Helper
You're right. I missed the "Each second". Of course, that is not Zeno's paradox because the rabbit is require to wait one second between hops and so clearly will never get to the lettuce.

#### FluxCapacitator

Whenever Zeno's Paradox is discussed, I always feel that I am missing something because it just seems intuitive to me that the chaser will never reach the chasee under the conditions given...is there some deeper complexity that I'm missing?

#### NateTG

Homework Helper
FluxCapacitator said:
Whenever Zeno's Paradox is discussed, I always feel that I am missing something because it just seems intuitive to me that the chaser will never reach the chasee under the conditions given...is there some deeper complexity that I'm missing?
Actually, in Zeno's paradox, the chaser catches up, and indeed passes, the chasee. The original question in this post relatively accurately describes the sort of situation that Zeno's paradox is conflating with constant velocity motion.

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