Solving Kinematics w/ Negative Acceleration: Meaning of Negative Root?

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Discussion Overview

The discussion revolves around a kinematics problem involving a particle moving along the x-axis with constant negative acceleration. Participants explore the implications of obtaining a negative root from the quadratic equation used to determine the time at which the particle reaches the origin.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of the negative root in the context of the problem, asking if it is extraneous as suggested by their teacher.
  • Another participant proposes that the negative solution represents the time when the particle was previously at the origin, while the positive solution indicates when it will next reach the origin.
  • A further participant seeks clarification on whether the particle crosses the origin and then moves backwards due to negative acceleration.
  • Another participant challenges the original problem's setup, suggesting that if the particle starts at -5 with a positive velocity and negative acceleration, both roots should be positive, indicating a potential error in the initial conditions presented.
  • A participant affirms that the explanation regarding the negative root is correct, referencing JBriggs's earlier response.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the negative root, with some supporting the idea that it reflects past motion while others contest the initial conditions leading to the roots. The discussion remains unresolved regarding the correctness of the initial problem setup and the interpretation of the roots.

Contextual Notes

There are potential limitations in the assumptions made about the initial conditions of the problem, particularly regarding the signs of velocity and acceleration, which may affect the interpretation of the roots.

oneplusone
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Suppose a particle is moving along the $x$ axis with velocity $v$. It starts at the point $-5$ and has constant negative acceleration $a$.
You need to find what time the particle reaches the origin.

My teacher solved this question, and used the kinematics equation Δx = v_0t+1/2 at^2

The quadratic had two roots, one of which was negative. What does the negative root mean? Is it extraneous (my teacher just said to ignore it)?

Thanks.
 
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The givens of the problem can be seen as specifying the position of the particle not only at all future times, but at all past times as well. The negative solution is the time when the particle was previously at the origin. The positive solution is when the particle will next be at the origin.
 
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So does this mean that the particle crosses the origin, and later the negative acceleration causes the particle to eventually go backwards, back to the origin?
 
oneplusone said:
So does this mean that the particle crosses the origin, and later the negative acceleration causes the particle to eventually go backwards, back to the origin?

Something is wrong with your signs in the original statement of your problem. If the particle started at -5 (left of the origin) with a positive velocity (right-moving) and a negative acceleration (being pulled to the left) both roots should have been positive, corresponding to exactly what you're describing here.

JBriggs's answer as to the meaning of a negative root in these problems is correct.
 

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