What Time Does the Particle Reach the Y-Axis?

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Homework Help Overview

The problem involves a particle moving in the xy-plane, with its position defined by parametric equations for x(t) and y(t) over the interval from 0 to π. The objective is to determine when the particle reaches the y-axis, which occurs when x(t) equals zero, and to find the speed and acceleration vectors at that time.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the difficulty of solving x(t) = 0 algebraically and suggest that a reasonable estimate for t could be sufficient. There is also mention of needing to find a t value within the specified interval, leading to questions about the validity of t = 0 as a solution.

Discussion Status

The discussion is ongoing, with participants exploring the possibility of estimating a solution for t while acknowledging the limitations of algebraic methods. Some guidance has been provided regarding the nature of the solution, but no consensus has been reached on the exact approach or outcome.

Contextual Notes

Participants note that the problem specifies t must be greater than 0, which influences their reasoning about potential solutions. There is also a mention of the teacher's claim that an exact solution exists, which adds to the complexity of the discussion.

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Homework Statement


A particle moves in the xy-plane so that its position at any time t, 0 =< t =< pi, is given by:
[tex]x(t) = \frac{t^2}{2}-ln(1+t)[/tex]
[tex]y(t) = 3sint[/tex]

-- At that time is the particle on the y-axis on the interval? Find the speed and acceleration vector of the particle at this time.


Homework Equations


None?


The Attempt at a Solution


I'm actually lost -- I believe that when x(t) = 0 it will be on the y-axis but I must be doing the algebra wrong I can't seem to get an exact solution...
 
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You can't solve it exactly algebraically. A well reasoned guess for a value of t such that x(t)=0 will give you the answer though.
 
Dick said:
You can't solve it exactly algebraically. A well reasoned guess for a value of t such that x(t)=0 will give you the answer though.

So my method is right up to the point I was at? I should mention I needed to find a t value such that 0 < t < pi, so I don't think 0 is the answer, I should have added that, sorry. Eye-balling the graph it's something like t=1.25 but my teacher claims that it is possible to get an exact solution, but not that you've confirmed it I really don't think there is...

Oh and, velocity is <x'(t), y'(t)> or (x'(t))i+(y'(t))j and acceleration is <x''(t), y''(t)> or (x''(t))i+(y''(t))j
 
What's wrong with t=0?
 
Dick said:
What's wrong with t=0?

Nothing I just forgot to add that the problem asks for a value of t > 0. I know it works but the problem isn't asking for it I don't think. My mistake I should have clarified:(
 
Last edited:
Your post does say t>=0. You've eyeballed the t>0 root correctly. But there's no way you can solve for that one in terms of elementary functions.
 

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