What is the total distance Achilles travels when catching the tortoise in Zeno's paradox?

Conditions

• $v_A = 2$
• Catch-up occurs at $t=2$

Reasoning, step by step

1. Identify the spatial series: $s_{\text{total}} = 2 + 1 + \frac{1}{2} + \frac{1}{4} + \cdots$.
2. Calculate the sum of this geometric series with first term 2 and ratio $1/2$.
3. Alternatively, multiply Achilles' speed ($v_A=2$) by the total time ($t=2$).
4. Result is 4.

Example

The script concludes: 'The series sums to a total time of 2 and an Achilles distance of 4.'

Common misconceptions

• Calculating the tortoise's distance instead (which would be $1 \times 2 = 2$ plus initial gap 2, totaling 4 from start, or 2 from its own start point depending on reference frame interpretation, but Achilles' run distance is clearly 4).
• Assuming the distance is infinite due to infinite steps.

Watch the explanation

Explore next

• Where is the tortoise located when caught?
• How does this match the kinematic equation $d=vt$?