Why is the slope of a V2 Vs. X Graph 2a?

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Discussion Overview

The discussion revolves around understanding why the slope of a graph plotting V² (final velocity squared) against position (X) is equal to 2a, where 'a' represents acceleration. Participants explore the underlying equations and relationships between velocity, time, and position in the context of kinematics.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on the equations that lead to the conclusion that the slope of the V² vs. X graph is 2a.
  • Another participant provides equations related to velocity and position, suggesting that if initial velocity (vi) is zero, the equation vf² = vi² + 2ax can be rearranged to show the slope of the V² vs. X graph.
  • A participant expresses confusion about the origin of the number 2 in the equation vf² = vi² + 2ax, questioning its derivation.
  • Further elaboration is provided by another participant, who attempts to derive the equation by multiplying the velocity equation vf = vi + at and manipulating it to show how the term 2a arises.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the derivation of the slope and the significance of the number 2 in the equation. There is no consensus on the clarity of the explanation provided, as some participants are still seeking answers while others attempt to clarify the derivation.

Contextual Notes

The discussion highlights potential gaps in understanding the manipulation of kinematic equations and the assumptions made regarding initial conditions, such as initial velocity being zero.

YungEggy
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I just need to know what equations and everything is used to explain why the slope of V2 Vs. X graph is 2a. I understand why V vs. T is a and X vs T2 is 1/2a.
Thanks!
 
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a = v/t (velocity vs. time)

x = (1/2) a t2 -> x / t2 = (1/2) a (x vs. t2 is a/2)

vf2 = vi2 + 2 a x

If vi2 = 0, you can rearrange the above equation to get the slope of the vf2 vs. x graph similar to the way you got the slope of the x / t2 graph.
 
But the only problem I have is where did the 2 come from? I understand how to arrange it to get the slope of a postion time graph.
 
YungEggy said:
But the only problem I have is where did the 2 come from?

Which 2? vf2 = vi2 + 2 a x This one?

Multiply the equation vf = vi + at with itself
vf2 = vi2 + a2t2 + 2atvi
vf2 = vi2 + 2a( vit + at2/2)
vf2 = vi2 + 2ax
 

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