$$\require{cancel}$$

6.3: Uniformly Accelerated Motion

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

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

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

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

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