Units Mismatch in Analytical Mechanics Textbook

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SUMMARY

The discussion centers on the terminal velocity equation for a spherical object as presented in an Analytical Mechanics textbook. The equation is given as vt=[(mg/c2)+(c1/2c2)2]1/2-c1/(2c2). Initially, there appears to be a units mismatch; however, upon analysis, it is confirmed that the units of c1 (Mass/Time) and c2 (Mass/Length) align correctly when substituted into the equation. This ensures that all terms in the equation are consistent in terms of units, validating the formulation.

PREREQUISITES
  • Understanding of terminal velocity in fluid dynamics
  • Familiarity with drag force equations, specifically Fdrag = c1 * v + c2 * v^2
  • Knowledge of dimensional analysis and unit consistency
  • Basic principles of Analytical Mechanics
NEXT STEPS
  • Study the derivation of the terminal velocity equation in fluid dynamics
  • Learn about the implications of linear vs. quadratic drag forces
  • Explore dimensional analysis techniques for verifying equations
  • Investigate the role of coefficients c1 and c2 in different fluid scenarios
USEFUL FOR

Students of physics, educators in mechanics, and anyone involved in fluid dynamics or analytical mechanics will benefit from this discussion.

shanepitts
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In my Analytical Mechanics textbook it states that the terminal velocity of a spherical object, whilst taking the linear and quadratic terms into consideration, is

vt=[(mg/c2)+(c1/2c2)2]1/2-c1/(2c2)

The units don't seem to match here. Is this normal?

This is not a homework inquiry, it for personal reviewing.
 
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Why don't they match? I think your c1 and c2 are from the expression Fdrag = c1 * v + c2 * v^2. Then the units of c1 are Mass/Time, and the units of c2 are Mass/Length. With these units, the units of c1/c2 is Length/Time, and the units of mg/c2 are Length^2/Time^2, and everything matches.
 
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