Question

The Greek philosopher Zeno of Elea (who lived about 450 B.c.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued The slower when running will never be overtaken by the quicker: for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. In other words, by giving the tortoise a head start, Achilles will never overtake the tortoise because every time Achilles reaches the point where the tortoise was, the tortoise has moved ahead. Resolve this paradox by assuming that Achilles gives the tortoise a 1 -mi head start and runs \(5 \mathrm{mi} / \mathrm{hr}\) to the tortoise's \(1 \mathrm{mi} / \mathrm{hr}\). How far does Achilles run before he overtakes the tortoise, and how long does it take?

Short answer

Answer: Achilles runs 1.25 miles before overtaking the tortoise, and it takes him 0.25 hours (or 15 minutes) to do so.

Step-by-step solution

Define variables and parameters

Let's define the following variables: - D: distance run by Achilles before he overtakes the tortoise (in miles) - T: time taken for Achilles to overtake the tortoise (in hours) The following parameters are given: - A\(_s\): speed of Achilles = \(5 \, \text{mi/hr}\) - T\(_s\): speed of the tortoise = \(1 \, \text{mi/hr}\) - H\(_s\): head start of the tortoise = \(1 \, \text{mi}\)

Calculate the relative speed

The relative speed, R\(_s\), is the difference in their speeds: R\(_s\) = A\(_s\) - T\(_s\) = \(5 \, \text{mi/hr}\) - \(1 \, \text{mi/hr}\) = \(4 \, \text{mi/hr}\)

Calculate the time it takes for Achilles to overtake the tortoise

Since relative speed is the rate at which Achilles gains distance on the tortoise, we can calculate the time it takes for him to cover the head start distance of \(1 \, \text{mi}\). Let T be the time in hours for Achilles to overtake the tortoise. Time = Distance / Relative Speed T = H\(_s\) / R\(_s\) = \(1 \, \text{mi}\) / \(4 \, \text{mi/hr}\) = \(0.25 \, \text{hrs}\)

Calculate the distance Achilles runs before overtaking the tortoise

Now we can find the distance Achilles runs by multiplying the time taken by Achilles' speed (A\(_s\)): D = T * A\(_s\) D = \(0.25 \, \text{hrs} \times 5 \, \text{mi/hr}\) = \(1.25 \, \text{mi}\) Hence, Achilles runs \(1.25\) miles before he overtakes the tortoise, and it takes him 0.25 hours (or 15 minutes) to do so.

Key concepts

Calculus Problems

Calculus, the mathematical study of continuous change, is a branch that deals with understanding the properties and applications of derivatives and integrals. Problems in calculus often involve finding the rate at which something changes (derivatives) or the total amount of change over an interval (integrals).

In the context of Zeno's paradoxes, calculus isn't used directly to solve the paradox but it provides a framework for understanding motion and limits. Specifically, Zeno's paradox can be considered as a problem of infinite series and sequences, which are fundamental concepts in calculus. The paradox suggests that an infinite number of steps must be completed, which seems impossible, but calculus allows the summing of an infinite series to reach a finite limit. This is the subtle connection between Zeno's paradox and problems in calculus.

Relative Speed

Relative speed is a concept used frequently in physics and mathematics to understand how fast one object is moving in relation to another. It is particularly useful in problems involving two or more entities moving towards or away from each other, like vehicles on a road or, as in Zeno's paradox, a race between Achilles and the tortoise.

The formula to calculate relative speed, when two entities are moving in the same direction, is simply the difference between their speeds. In the Achilles-tortoise scenario, since they are moving along the same path, Achilles' relative speed to the tortoise is his speed minus the tortoise's speed. It is the speed at which the distance between them changes and is crucial to solving problems where time and distance relationships for overtaking or meeting are to be found.

Distance-Time Relationship

The distance-time relationship is fundamental in understanding motion. It tells us how distance covered changes over time. This relationship is often represented as a graph where time is on the x-axis and distance on the y-axis, and the slope of the line represents the speed of the moving object.

In our given problem, the relationship tells us how far Achilles must run to catch up to the tortoise and how this distance changes as time passes. The formula for calculating this is quite straightforward: Distance equals speed multiplied by time. Importantly, in the context of Zeno's paradox, this relationship helps clear the confusion: each segment Achilles runs is smaller and takes less time than the previous one, converging to the point where he overtakes the tortoise.