Puzzle #7 – Resistive Forces

A particle of mass m is acted on by a force

F=\lambda e^{-\frac{x}{\gamma}}

where \lambda and \gamma are positive constants, and x is the particle’s displacement from the origin.

Question: By using the chain rule, show that this equation of motion is exactly equivalent to one of the form

F=\lambda-\rho v^2

and deduce an equivalence for \rho. How might you describe the two forces at work?

Extension: Given that the particle starts from rest at the origin, show that the particle’s position x after time t is given by

x=2\gamma\ln\bigg(\cosh\Big[t\sqrt{\frac{\lambda}{2m\gamma}}\Big]\bigg)

or an equivalent, expressed in terms of \rho.

Hint: The chain rule provides the following relation:

\displaystyle \frac{dv}{dt}=\frac{dv}{dx}\cdot\frac{dx}{dt}

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One thought on “Puzzle #7 – Resistive Forces

  1. Pingback: Puzzle #8 – Unappealing Puzzle | Conversation of Momentum

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