Find the velocity vector for the particle at any time

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Discussion Overview

The discussion revolves around finding the velocity vector of a particle moving along a specified path, as well as evaluating an integral expression to determine the distance traveled by the particle over a specific time interval. The scope includes mathematical reasoning and conceptual clarification regarding velocity and position vectors.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Homework-related

Main Points Raised

  • One participant seeks to find the velocity vector for a particle defined by the parametric equations x(t)=3cos(pi*t) and y(t)=5sin(pi*t).
  • Another participant suggests that the velocity vector is related to the concept of a position vector and how positions correlate with velocities.
  • A participant expresses uncertainty about the relationship between velocity and slope.
  • It is noted that the position vector describes the particle's location in space, which requires a vector quantity for accurate representation.
  • One participant proposes that the integral expression for distance traveled resembles an arclength calculation, indicating a method to derive the necessary components for evaluation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concepts of velocity and position vectors, with some agreement on the relationship between position and velocity, but no consensus on the specific calculations or interpretations of the velocity vector.

Contextual Notes

There are unresolved assumptions regarding the definitions of velocity and position vectors, as well as the mathematical steps required to derive the velocity vector and evaluate the integral for distance.

Who May Find This Useful

This discussion may be useful for students or individuals interested in understanding the relationship between position and velocity in the context of parametric equations and integral calculus.

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During the time period t=0 to t=6 seconds, a particle moves along the path given by x(t)=3cos(pi*t) and y(t)=5sin(pi*t)

1. Find the velocity vector for the particle at any time t.
2. Write and evaluate an integral expression, in terms of sine and cosine, that gives distance the particle travels from time t=1.25 to t=1.75.

Not quite sure what a velocity vector is...

For the second, it looks like an arclength so it would be Integral from 1.25 to 1.75 Sqrt((dy/dt)2 + (dx/dt)^2), so just derive the given (3cos(pi*t), etc. and plug in?
 
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1. Do you know the concept "position vector", and how positions are related to velocities?
2. That's right.
 
Does it have something to do with slope?
 
The position vector gives the point in space where the particle is at a given time.
Since, in general, that point in space has three coordinates (x,y,z), the particle's position must be described with a vector quantity, not a single number.
Hope this helps..
 

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