Terminal velocity and drag force in one dimention

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SUMMARY

The discussion centers on the behavior of a particle's velocity in one-dimensional motion under the influence of drag force, described by the equation mv' + bv = F(t). It concludes that as time approaches infinity, the velocity v approaches F(c)/m, where c is a constant. The theorem referenced indicates that under certain conditions, the integral of the product of two continuous functions can be evaluated, supporting the findings regarding terminal velocity. The participant acknowledges the need for clarification on determining the value of c in practical applications.

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear equations.
  • Familiarity with the concept of terminal velocity in physics.
  • Knowledge of continuous functions and definite integrals.
  • Basic grasp of drag force and its mathematical representation.
NEXT STEPS
  • Study the derivation of terminal velocity in various drag force scenarios.
  • Explore numerical methods for solving first-order differential equations.
  • Learn about the implications of the constant c in different physical contexts.
  • Investigate the application of the theorem in real-world problems involving continuous functions.
USEFUL FOR

Students and professionals in physics, particularly those focusing on mechanics and fluid dynamics, as well as mathematicians interested in differential equations and their applications.

Lucien1011
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In one dimensional cases, will the velocity of a particle tend to the terminal velocity unregardless of any combinations of forces?
I try to investigate this equation: mv'+bv=F(t)
Using the mathematics theorem at the botton, I found that v --> F(c)/m as t tends to infinity. (where c is some constant)

[Thm: if w(x) and u(x) are continuous functions and u(x)>=0, then for a<=x<=b, then {w(x)u(x)}:b,a=w(c)*{u(x)}:b,a for some c lies between a and b]
the notation {f(x)}:b,a represents the definite integral from a to b with repect to x. Sorry for the unusual notation, as I don't know how to type the integral.

Sorry for the poor presentation too. I intended to write the result on a piece of paper and scan it into the computer but my scanner is out of order now.
 
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Looks fine to me if the theorem is correct. How are you going to determine the value of c?
 
matematikawan said:
Looks fine to me if the theorem is correct. How are you going to determine the value of c?
Oh yeah. I neglect that the upper and lower limit in the thm should be constants.
 
Last edited:

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