Total distance traveled by particle

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Homework Help Overview

The discussion revolves around determining the total distance traveled by a particle using its position function, s(t) = 2t^3 - 21t^2 + 60t, over a specified time interval. Participants are exploring different methods for calculating distance and the implications of each approach.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to compare two methods for calculating total distance: evaluating the position function at discrete time points versus differentiating the function to find critical points. Some participants question the necessity and implications of setting the derivative to zero.

Discussion Status

The discussion is active, with participants examining the standard approach of differentiation and its relevance to the problem. There is an exploration of alternative methods and their potential outcomes, but no consensus has been reached regarding the best approach.

Contextual Notes

Participants are considering the implications of using different functions and how those might affect the calculations, particularly in the context of the original poster's example and a hypothetical function, s(t) = 4t^2 - 4t + 1.

BrownianMan
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If s(t) = 2t^3 - 21t^2 + 60t is the position function of a particle moving in a straight line, would you be able to find its total distance traveled in, say 3 seconds, by finding s(0), s(1), s(2), s(3), and calculating the absolute value between each of them and then summing those values, as opposed to differentiating the function first, setting the derivative to 0, and solving for t?

Would you get the same answer?
 
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What do you hope to gain by setting ds/dt to 0?
 
That's the standard way of solving such a problem - or at least how we've been taught. You differentiate to find the velocity, then set velocity to 0 to find the intervals when the particle is moving in a positive and in a negative direction.
 
OK, just making sure you understand. What would happen in both cases if the function was say, s(t) = 4t2-4t+1?
 

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