How Do You Calculate Terminal Velocity Involving Drag?

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Discussion Overview

The discussion revolves around deriving the equation for the terminal velocity of a falling body, specifically focusing on the role of drag that is proportional to speed. Participants explore the integration steps involved in the derivation and address a potential mistake in handling signs during the integration process.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents the equation of motion for a falling body, incorporating drag, and outlines their steps for deriving terminal velocity.
  • Another participant suggests that dividing the equation by -b initially might lead to a correct result, indicating a possible alternative approach.
  • A third participant points out a missed minus sign during integration, referencing the chain rule.
  • A later reply acknowledges the oversight and expresses gratitude for the correction.

Areas of Agreement / Disagreement

Participants appear to agree on the general approach to the problem but have differing views on the handling of signs during integration. The discussion remains unresolved regarding the exact steps leading to the final equation.

Contextual Notes

There is a potential limitation regarding the assumptions made about the drag force and its proportionality to speed, as well as the integration steps that may depend on specific mathematical interpretations.

spacetimedude
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Hello PF,
I have once simple (well, not so simple for me) question.

I'm trying to derive an equation for the velocity of a falling body with accordance to terminal velocity.

The equation incorporates drag proportional to the speed.

m*dv/dt=mg-bv

and

mg/b=terminal velocity vt

So the steps I took were:

m*dv/dt+bv=mg

(m/b)*(dv/dt)+v=vt

dv/dt=(b/m)(vt-v)

dv/(vt-v)=(b/m)dt

Integrating both sides would give
ln[(vt-v)/(vt)]=(b/m)t

But the textbook says that I'm supposed to get negative (b/m)t on the left side.

Have I made a mistake on the integration part?

Any help will be deeply appreciated.
 
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hmm. I seem to have gotten the answer if I just divided the entire equation by -b in the beginning without bringing the bv to the left side. Have I made a mistake on the integration part?
 
You just missed the minus sign while integrating (chain rule)
 
pshh. I can't believe I missed that. Thanks so much king vitamin!
 

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