Prove c(t) is a Straight Line or Point w/ Zero Acceleration

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Homework Help Overview

The problem involves proving that a path \( c \) in \( \mathbb{R}^3 \) with zero acceleration must be either a straight line or a single point. The context combines elements of physics and mathematics, particularly focusing on the implications of zero acceleration on the motion described by the path.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the implications of zero acceleration, noting that it suggests constant velocity. There is an exploration of integrating zero to demonstrate that velocity remains constant, leading to the conclusion that the path could be linear. Questions arise about how to mathematically prove the constancy of velocity and whether a path could also represent a single point.

Discussion Status

Some participants have provided insights into the integration of zero and its implications for the path's equation. There is recognition that if the constant \( C \) is zero, the path could indeed represent a single point. Multiple interpretations of the problem are being explored, particularly regarding the definitions and implications of acceleration in this context.

Contextual Notes

There is some confusion regarding the terminology used in the problem statement, with one participant expressing frustration over the mixing of physics and mathematics concepts. This indicates a potential area of misunderstanding that may affect the discussion.

doppelganger007
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Homework Statement


Let c be a path in R^3 with zero acceleration. Prove that c is a straight line or a point.


Homework Equations


F(c(t)) = ma(t)
a(t) = c''(t)


The Attempt at a Solution


so i know that since the acceleration is zero, the velocity must be constant, and when you integrate a constant, you get a straight line...but how to I prove mathematically that the velocity is constant, because you can't integrate 0dt, as far as I know?
 
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The indefinite integral, i.e, the anti-derivative of 0 is, indeed, a constant; that is we have:
[tex]\int{0}dx=C[/tex]
 
oh ok, so if I integrate that again I get that c(t) = Ct + D, which fits the general equation for a line

but then, does that also prove that c(t) could just be a single point?
 
Indeed, since big C could be..0!
 
oh. duh!
gratzie
 
doppelganger007 said:

Homework Statement


Let c be a path in R^3 with zero acceleration. Prove that c is a straight line or a point.


Homework Equations


F(c(t)) = ma(t)
a(t) = c''(t)


The Attempt at a Solution


so i know that since the acceleration is zero, the velocity must be constant, and when you integrate a constant, you get a straight line...but how to I prove mathematically that the velocity is constant, because you can't integrate 0dt, as far as I know?
Damn, I hate mixed "physics" and "mathematics" problems! You or whoever set this problem, should know that a "path" DOES NOT HAVE an "acceleration". I expect this problem should be "find the equation of motion of a particle whose trajectory is a given path in R3 with acceleration 0. Show that the path is either a straight line or a point". Then you would begin with [itex]\vec{a}= d\vec{v}/dt=[/itex] and go from there.
 
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The easiest way would have been to recognise that acceleration is a vector quantity, it is affected both by direction or magnitude. No acceleration, no change in direction, which means constant gradient. Simple as that.
 

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