If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

Question

There is a ball that starts at point A on a line and moves toward point B.

Every second, it moves half of the distance left, but never stops moving:

Etc.

Would the ball ever reach point B?

In one perspective, you could argue that since the ball is perpetually moving, it will reach the end sometime.

You could also argue that the ball would never reach the end because it would always be moving half the distance left, never the full.

Which is correct? To me, I think that if something is constantly advancing, it would eventually get from A to B. Does it work that way? If it does, at what time would the ball reach the end? Thanks for the help.

Answer 1

After any finite amount of time, the ball will not be at the point $B$. In fact, after $n$ seconds, the ball will travel $1 - \frac{1}{2^n}$ of the full distance between $A$ and $B$. This is because after one seconds, the ball travels a distance of $\frac12$, after $2$ seconds, it travels $\frac12 + \frac 34$, and after $n$ seconds, it travels

$$\sum_{i=1}^n \frac1{2^i} = 1 - \frac{1}{2^n}$$

So, what is wrong with your logic? You say "if something is constantly advancing, it will eventually get from $A$ to $B$", which is false. Your example shows quite clearly that even if something is constantly advancing, if the amount it is advancing by is decreasing, it may never reach its destination.

Answer 2

If every second the speed $v=v(t)$ is halves then $v$ satisfies the equation $$ v(t)=k\cdot 2^{-t}=k\,\mathrm{e}^{-t\log 2}=v(0)\,\mathrm{e}^{-t\log 2}. $$ Thus $$ \int_0^\infty v(t)\,dt=\frac{v(0)}{\log 2}. $$ This means that, it can cover at most $\dfrac{v(0)}{\log 2}$

Answer 3

This is similar to Zeno's Paradox (Achilles and the tortoise).

If you're going to consider infinite terms (progressively getting smaller) culminating to a finite answer, then you'll simply have to sum up a geometric series, or use calculus.

More details: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Proposed_solutions

It reminds of this joke:

A mathematician and a physicist agree to a psychological experiment. The mathematician is put in a chair in a large empty room and a delicious plate of food is placed on a table at the other end of the room. The psychologist explains, "You are to remain in your chair. Every five minutes, I will move your chair to a position halfway between its current location and the food on the table." The mathematician looks at the psychologist in disgust. "What? I'm not going to go through this. You know I'll never reach the table!" And he gets up and storms out. The psychologist makes a note on his clipboard and ushers the physicist in. He explains the situation, and the physicist's eyes light up and he starts drooling. The psychologist is a bit confused. "Don't you realize that you'll never reach the food?" The physicist smiles and replied, "Of course! But I'll get close enough for all practical purposes!"