How can the ball reach the wall when it always has to travel halfway?

Question

If I throw a ball at the wall, when it has travelled halfway, it still has half the distance to travel. As it continues, the fraction left to travel continues i.e. one quarter to go, one eighth to go, one sixteenth to go, etc. since the denominator of the distance continues to double... I am thinking mathematicaly speaking, the ball never reaches the wall.

Answer 1

Mathematically speaking: $$\sum_{n=1}^\infty\frac{1}{2^n}=1$$ You're just restating Zeno's, *ahem*, "paradox" (Wikipedia link).

Answer 2

The ball reaches the wall since the time interval required to go half of the remaining distance to the wall decreases by a factor of $2$ with each successive step (assuming constant speed throughout the journey).

So, for the first step, the ball travels $1/2$ the distance to the wall and takes a time, say $t_1$.

For the second step, the ball travels $1/2$ the remaining distance to the wall ($1/4$ of the original distance), taking $\frac12 t_1$ to get there.

Continuing, we have the total time that it takes to reach the wall is

$$\sum_{n=0}^{\infty}\frac{t_1}{2^n}=2t_1$$

which is certainly less than "forver."