The Tortoise and The Hare --- Zeno Phenomenon (2)

Lynn G included in category Calculus

2025-09-26 2025-09-26 715 words 4 minutes

Contents

1 Story Time Twist

The other night, I was reading The Tortoise and the Hare with my kid. You know the ending — the rabbit takes a nap, the tortoise keeps going, and… yep, the tortoise wins. Classic.

But when we finished the story — yeah, you caught me — I just couldn’t resist adding a twist: " What if the very next day… they race again?"

This time, the rabbit wants to look “fair.” He smirks and says: “Tortoise, you’re way too slow. To make things exciting, I’ll give you a 100-meter head start!”

The race begins.

Whoooosh! The rabbit blasts off and quickly covers that 100 meters. But when he hits the tortoise’s starting line — guess what? the tortoise has crawled 10 meters farther.

The rabbit keeps running fast and quickly caught up the 10 meters … only to see the tortoise has moved ahead by 1 more meter.

The rabbit tries again. He runs that 1 meter in almost no time… but the tortoise has gone 0.1 meter farther…

And so it goes — every time the rabbit gets close, the tortoise is just a little farther ahead!

At this point, the kid’s eyes went wide: “Wait, what? Does that mean the rabbit can never catch the tortoise?”

Is that possible? Or is there a deeper trick hiding here?

2 Intuitive Explanation

I gave a little mysterious smile,

“Hey, remember when you raced with your friend Andy last week? He dashed ahead quite a long way while you weren’t looking. And… what happened then?”

My kid jumped in right away, full of confidence:

“I caught him! Because I was much faster — and I even passed him!”

Then he slapped his forehead like he’d just solved the biggest puzzle:

“OHHHH—so the rabbit can catch the tortoise after all!”

I nodded, with a gentle grin:

" Yep! The faster one always catches the slower one in an amount of time… even if it feels like you’re chasing again, and again, and again."

3 The “Trap” of Zeno Phenomenon

In the story, every time the rabbit gets to where the tortoise was, the tortoise has already moved a tiny bit ahead. It feels like the rabbit is always just a little too late, and can never catch up!

This is the very ’trap’ of Zeno’s paradox: mixing up infinitely many chasing steps with an infinitely long amount of time.

Up next, let’s bring in the idea of limits to reveal the surprise: even with infinitely many steps, the total time is actually finite!

Alight, fun time for us big kids – ready, go!

4 Math Explanation

4.1 Setup

Tortoise has a head start distance $D$.

Tortoise speed: $v_T$.

Rabbit speed: $v_A$, with $v_A > v_T$.

We break the chase into infinitely many segments: each time the rabbit runs to where the tortoise was, the tortoise moves further.

4.2 Distances / gaps

Let $d_1 = D.$

Time taken by Achilles to cover $d_1$ is

t_1 = \frac{d_1}{v_A} = \frac{D}{v_A}.

During that time, tortoise advances:

\Delta_1 = t_1 \cdot v_T = \frac{D}{v_A} \cdot v_T.

Thus the remaining gap this time becomes

d_2 = \Delta_1 .

By the same logic,

t_2 = \frac{d_2}{v_A} = \frac{D}{v_A} \cdot \frac{v_T}{v_A}.

Then the tortoise advances again, gap shrinks, etc. We get the general term

\quad t_n = \frac{d_n}{v_A} = \frac{D}{v_A} \left(\frac{v_T}{v_A}\right)^{\,n-1}.

4.3 Time sum (infinite series)

So the total time Achilles needs to catch the tortoise is

T = \sum_{n=1}^\infty t_n = \frac{D}{v_A} \sum_{n=0}^\infty \left(\frac{v_T}{v_A}\right)^n.

Since $\tfrac{v_T}{v_A} < 1$,the series converges (geometric series):

\sum_{n=0}^\infty \left(\frac{v_T}{v_A}\right)^n = \frac{1}{1 - \tfrac{v_T}{v_A}} = \frac{v_A}{v_A - v_T}.

Hence

T = \frac{D}{v_A} \cdot \frac{v_A}{v_A - v_T} = \frac{D}{v_A - v_T}.

That is a finite positive time. After that time $T$, the rabbit catches up.

5 Closing

Just like the race between the Tortoise and the Hare, or that treasure hunt we talked about earlier with steady speed, the idea of infinitely many events fitting into a limited time is what we call a Zeno behavior.

Think of dropping a little ball: Boing! it bounces high the first time, then lower, then even lower… The trick to handling problems like these is:

instead of getting stuck on the thought of ‘forever’, we assume a definite stopping time, and then calculate the final result after all those infinitely tiny bounces.”

Bam! >_<