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# 6: Motion in a Resisting Medium

• • Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

In studying the motion of a body in a resisting medium, we assume that the resistive force on a body, and hence its deceleration, is some function of its speed. Such resistive forces are not generally conservative, and kinetic energy is usually dissipated as heat. For simple theoretical studies one can assume a simple force law, such as the resistive force is proportional to the speed, or to the square of the speed, or to some function that we can conveniently handle mathematically. For slow, laminar, nonturbulent motion through a viscous fluid, the resistance is indeed simply proportional to the speed, as can be shown at least by dimensional arguments. One thinks, for example, of Stokes's Law for the motion of a sphere through a viscous fluid. For faster motion, when laminar flow breaks up and the flow becomes turbulent, a resistive force that is proportional to the square of the speed may represent the actual physical situation better.

• 6.1: Introduction
In studying the motion of a body in a resisting medium, we assume that the resistive force on a body, and hence its deceleration, is some function of its speed. Such resistive forces are not generally conservative, and kinetic energy is usually dissipated as heat. For simple theoretical studies one can assume a simple force law, such as the resistive force is proportional to the speed, or to the square of the speed, or to some function that we can conveniently handle mathematically.
• 6.2: Uniformly Accelerated Motion
Before studying motion in a resisting medium, a brief review of uniformly accelerating motion might be in order. That is, motion in which the resistance is zero. Any formulas that we develop for motion in a resisting medium must go to the formulas for uniformly accelerated motion as the resistance approaches zero.
• 6.3: Uniformly Accelerated Motion
Here we consider problems where the only force on a body is a resistive force that is proportional to its speed.
• 6.4: Motion in which the Resistance is Proportional to the Square of the Speed
Here we consider problems where the only force on a body is a resistive force that is proportional to the square of its speed. There are not really any new principles; it is just a matter of practice with slightly more difficult integrals.

Thumbnail: Laminar and turbulent water flow over the hull of a submarine. As the relative velocity of the water increases turbulence occurs. (Public Domain; US Navy).