91Ó°ÊÓ

The Greek philosopher Zeno of Elea (who lived about 450 s.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?

Step by step solution

01

Set up the equation for relative distance traveled

Let x be the distance that Achilles runs before he overtakes the tortoise. During the time it takes Achilles to run that distance, the tortoise will have traveled a distance that is (x-1) since the tortoise has a 1-mile head start.

02

Determine the relative time

Since we are trying to find when Achilles catches up to the tortoise, we can set up the equation: Achilles' travel time = Tortoise's travel time The travel time is equal to the distance traveled divided by the speed. For Achilles: $$\frac{x}{5}$$ For the tortoise, the distance traveled is (x-1): $$\frac{(x-1)}{1}$$ Now we can set up the equation: $$\frac{x}{5} = \frac{(x-1)}{1}$$

03

Solve for x (the distance Achilles runs)

To solve for x, we need to multiply both sides of the equation by 5: $$x = 5(x-1)$$ Now, distribute the 5: $$x = 5x - 5$$ Next, subtract 4x from both sides: $$-4x = -5$$ Finally, divide by -4 to find x: $$x = \frac{5}{4} = 1.25$$ So, Achilles runs 1.25 miles before he overtakes the tortoise.

04

Calculate the time it takes for Achilles to overtake the tortoise

To find the time it takes, we can use the travel time formula for Achilles with the distance of 1.25 miles: Travel time = $$\frac{\text{distance}}{\text{speed}}$$ $$\text{Time} = \frac{1.25 \text{ miles}}{5\,\text{mi} \cdot \text{hr}^{-1}} = \frac{1}{4}\,h$$ Converting this to minutes: $$\frac{1}{4} \times 60 \,\text{min} = 15\,\text{min}$$ So, it takes Achilles 15 minutes to overtake the tortoise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Achilles and Tortoise Paradox

Zeno's famous Achilles and Tortoise Paradox presents a fascinating exploration into motion and time. It was derived by the Greek philosopher, Zeno of Elea. In this paradox, Achilles, a swift warrior, is in a race with a slower-moving tortoise, which is given a head start. Zeno suggests that since Achilles has to reach where the tortoise initially was, and every time Achilles reaches that point, the tortoise has already moved forward. It seems impossible for Achilles to ever catch up. This paradox challenges our understanding of motion and infinity, as it implies that motion consists of infinite steps within finite time. This thought experiment invites us to think deeply about how motion and time interrelate.

Relative Motion

Relative motion helps us understand how two objects move concerning each other. In solving Zeno’s paradox, the concept of relative motion becomes crucial. We look at how the speed of Achilles directly compares to that of the tortoise. For instance, although Achilles runs five times faster, we are interested in how and when he catches up. Relative motion allows us to break down and visualize the movement of one object from another's perspective, providing clarity in scenarios like races. Using relative motion, we can strip away the complexity introduced by each object’s movement through space and instead focus on distance and speed differences.

Solving Equations

Solving equations underpins the resolution of Zeno’s paradox. Once we set up our problem with variables and relationships, we can translate the verbal scenario into mathematical expressions. By calculating, we identify when Achilles meets the tortoise. The core equation to solve is:

• When Achilles' travel time equals the tortoise's.

This leads us to \[ \frac{x}{5} = \frac{x-1}{1} \]. As we solve, it simplifies to finding that distance where their times match. Solving relies on algebraic manipulation to express a scenario in terms we can analyze and understand deeply.

Distance-Time Relationship

The distance-time relationship is a pivotal part of solving Zeno’s paradox. It ties together how far an object travels, at what speed, and how long it takes. For Achilles and the tortoise, we calculate how long it takes Achilles to cover 1.25 miles at his speed of 5 mi/hr. The time (\( t \) ) it takes for Achilles to overtake is derived from the formula:

• \[ t = \frac{1.25}{5} = 0.25 \text{ hours} \]

This equates to 15 minutes. Understanding this concept helps in predicting and analyzing motion across various scenarios. By applying these principles, we can accurately depict how motion unfolds over time and distance.