Trouble Deriving a Constant Acceleration Formula

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Discussion Overview

The discussion revolves around the derivation of a formula for constant acceleration in the context of a physics problem involving a car crossing a 30 m-wide intersection while accelerating from rest. Participants explore the correct application of kinematic equations and the derivation process, focusing on the participant's misunderstanding of the formulas and their application.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion over their derivation of the formula and seeks clarification on their mistakes, specifically in the context of using the kinematic equations.
  • Another participant claims the original derivation is incorrect due to the assumption that velocity is displacement divided by time, which does not hold for non-constant velocity scenarios.
  • A third participant points out that the algebra in the participant's equations is incorrect, suggesting that the correct approach involves integration, which may not be familiar to the original poster.
  • In contrast, a fourth participant proposes an alternative method that avoids integration by using the concept of average velocity in constant acceleration scenarios, providing a step-by-step derivation of the distance formula.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach to derive the formula. There are competing views on the necessity of integration and the validity of the original derivation, indicating ongoing disagreement and uncertainty in the discussion.

Contextual Notes

Limitations include potential misunderstandings of the kinematic equations, the assumption of constant acceleration, and the algebraic steps taken by the original poster. The discussion does not resolve these issues definitively.

berenmacbowma
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Q: How long does it take a car to cross a 30 m-wide intersection after the light turns green, if it accelerates from rest at a constant 2.00 m/s^2?

Attempt: V=V0 + at; x-x0/t=v0+at; x-x0=v0+at^2; x=v0+x0+at^2; 30=0+0+2t^2; 15=t^2.

The square root of 15 ended up being my final answer, when the real answer was closer to 5.48 seconds. I just started physics, so I need to know what I personally did wrong when USING the formula I attempted to use. Giving me a different formula won't be helpful, because it won't tell me what I did wrong, and because my book has a whole chart of formulas I could refer to. If someone could explain to me how I derived that formula incorrectly, it would be a load of help. Thank you!
 
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You derived the formula incorrectly by assuming that velocity at time t is displacement/time. That it is not true when velocity is not constant. To compound the problem, your algebra is wrong in equation 3, but it doesn't matter since equation 2 is wrong.The kinematic equations are best derived form the calculus, check out a good website for this. You might want to refer to the chart of formulas in the book, and you will discover that none of them are the same as the one you derived.
 
berenmacbowma said:
Q: How long does it take a car to cross a 30 m-wide intersection after the light turns green, if it accelerates from rest at a constant 2.00 m/s^2?

Attempt: V=V0 + at; x-x0/t=v0+at

Right there; that does not follow. The only way you're going to be able to make the correct step is through integration, which if you're not familiar with isn't going to be possible at all.
 
You can avoid integration by noting that in a constant acceleration situation, average velocity for any time period equals half the sum of the starting and ending velocities.

average veloctiy = av = 1/2 (v0 + v1)
v1 = v0 + at
av = 1/2 (v0 + (v0 + at))
av = 1/2 (2v0 + at)
av = v0 + 1/2 a t

then distance equals x0 + average velocity x time:

x = x0 + av t = x0 + (v0 + 1/2 a t) t
x = x0 + v0 t + 1/2 a t2
 

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