Find the Equation of a Line (not simple)

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Discussion Overview

The discussion revolves around a problem involving the motion of two points on a Cartesian plane, where one point moves along the y-axis and the other moves towards it on the x-axis. Participants explore how to derive the equation of the path traced by the second point, considering both simplified and complex versions of the problem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario where Point 1 moves vertically and Point 2 moves horizontally towards it, seeking guidance on how to approach finding the equation of the line traced by Point 2.
  • Another participant suggests that the path traced by Point 2 resembles a tractrix, proposing that analytic expressions for arc length might lead to a solution.
  • A participant expresses uncertainty about the original problem's complexity and mentions using MATLAB for approximations, indicating a preference for a closed-form solution.
  • There is a discussion about the terminology used, with one participant suggesting that the path should be referred to as a curve rather than a line.
  • Some participants question the concept of constant velocity being periodic, leading to clarifications about the distinction between speed and velocity in the context of the problem.
  • Further clarification is provided regarding the motion of Point 1, with one participant describing it as oscillatory, while another relates it to circular motion and constant angular frequency.
  • Hints are provided regarding the mathematical representation of one particle moving directly at another and the need to formulate the motion of Particle 1 for further analysis.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement, particularly regarding the terminology of the path and the interpretation of motion. There is no consensus on the best approach to solve the problem, and various models and interpretations are presented.

Contextual Notes

Some participants express confusion about the definitions of velocity and speed, indicating potential limitations in understanding the problem's dynamics. The discussion also reflects varying levels of mathematical competence among participants.

Who May Find This Useful

This discussion may be useful for students and enthusiasts interested in mathematical modeling, dynamics, and the application of calculus to motion problems.

Samky
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TL;DR
Old puzzle -- I need help on how to approach
An old puzzle that was given to me long ago. I don't know how to approach this problem, any help is appreciated. I'm removing some specifics because I don't want it solved for me, I want to know how to approach it. Feel free to solve a simplified version if that helps you explain it to me.

Imagine two points on a standard Cartesian plane. Point one at the origin, and point 2 somewhere on the x axis. Now:

1) Point 1 moves along the y-axis at constant velocity and with period T

2) Point 2 moves at constant velocity and always directly at point 1

3) The velocities and period are such that the two points will meet at the origin

By tracing the path of point 2, we have a line.

Find the equation of this line.

----

Simplified version example:

Point 2 starts at (10, 0). Point 1 moves at 10 meters per second. Point 1 moves up 10 meters then down 10 meters. After two seconds they meet at (0, 0)
 
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You get a tractrix, or two combined tractrices in this case. It has analytic expressions for the arc length, so maybe you can find an analytic solution for the full problem. If not, numerical methods will find an approximation.
 
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Likes   Reactions: phinds
Thanks a lot, I'll look into that!

I wrote some MATLAB code to get a good approximation, so I have some graphs and numbers, but that's sort of cheating. I'm sure a mathematician could solve the original problem in closed form but I have no clue how to begin... but this should help me out, thanks again.
 
Samky said:
By tracing the path of point 2, we have a line.

It's probably a good idea to call this a curve and not a line.
 
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Samky said:
1) Point 1 moves along the y-axis at constant velocity and with period T
How can a velocity be both constant and periodic unless it is merely constant?
 
jbriggs444 said:
How can a velocity be both constant and periodic unless it is merely constant?
I'm confused by that too
 
Sorry, I mean speed. The magnitude doesn't change but the direction does. It's basically a one dimensional oscillation that creates a two dimensional oscillation.
 
Samky said:
Sorry, I mean speed. The magnitude doesn't change but the direction does. It's basically a one dimensional oscillation that creates a two dimensional oscillation.
I still can't picture the motion. An object can travel in a circle with constant speed, and if you take one component of its motion then you will have a one-dimensional oscillator. However, if you have a one-dimensional oscillator it must turn around 180 degrees at its maximum displacement. To do that, it must get to zero speed before turning around.

Do you mean it has a constant angular frequency?
 
It's an up/down motion with constant speed and alternating sign, like a frictionless ball bouncing between two walls.
 
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  • #10
mfb said:
It's an up/down motion with constant speed and alternating sign, like a frictionless ball bouncing between two walls.
Thank you!
 
  • #11
I've got an outline of a solution. Are you competent at vectors and derivatives?
 
  • #12
I'm student-level at vectors and derivatives. I've done basic vector calculus and all that. I'm confident I can solve simple problems involving these things.
 
  • #13
Here's the first two hints:
  • If a particle moves directly at another particle, mathematically how is this represented?
  • You'll also need to write out an equation for the motion of particle 1, which will be used later.
 

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