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6.3: Uniformly Accelerated Motion

Last updated

Aug 8, 2022
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- 6.2: Uniformly Accelerated Motion
- 6.3A: Resistive Force Only

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If the only force on a body is a resistive force that is proportional to its speed, the equation of motion is

$m\ddot{x}= -b\dot{x}. \nonumber$

One thinks, for example, of Stokes's equation for the laminar motion of a sphere through a viscous fluid, in which the resistive force is $6\pi\eta a\upsilon$ , where $\eta$ is the coefficient of dynamic viscosity. If we divide both sides of the equation by the mass $m$ , we obtain

$m\ddot{x}= -\gamma\dot{x}, \nonumber$

where $\gamma = \frac{b}{m}$ is the damping constant. It has dimension T^-1 and SI units s^-1.

Topic hierarchy

6.3A: Resistive Force Only
It is difficult to imagine a real situation in which the one and only force is a resistive force proportional to the speed. A body falling through the air won't do, because, in addition to the resistive force, there is the acceleration due to gravity. Perhaps we could imagine a puck sliding across the ice. The ice would have to be presumed to be completely frictionless, and the only force on the puck would be the resistance of the air.
6.3B: Body falling under gravity in a resisting medium, resistive force proportional to the speed
6.3C: Body thrown vertically upwards with initial speed v₀

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