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

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SUMMARY

The slope of the V2 vs. X graph is definitively established as 2a, derived from the kinematic equation vf2 = vi2 + 2ax. When the initial velocity (vi) is zero, this equation simplifies to vf2 = 2ax, clearly indicating that the slope corresponds to 2a. The discussion highlights the relationship between velocity, acceleration, and displacement, emphasizing the mathematical manipulation of kinematic equations to derive the slope.

PREREQUISITES
  • Understanding of kinematic equations, specifically vf2 = vi2 + 2ax.
  • Familiarity with basic algebraic manipulation of equations.
  • Knowledge of the relationship between velocity, acceleration, and time.
  • Concept of graphing relationships in physics, particularly V vs. T and X vs. T2 graphs.
NEXT STEPS
  • Study the derivation of kinematic equations in physics textbooks.
  • Learn about graphing techniques for velocity and acceleration in motion analysis.
  • Explore the implications of initial conditions on kinematic equations.
  • Investigate the relationship between acceleration and displacement through practical experiments.
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Students of physics, educators teaching kinematics, and anyone interested in understanding the mathematical relationships in motion analysis.

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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