# Math Help - The motion of a body falling in a viscous medium differential equation help (:

1. ## The motion of a body falling in a viscous medium differential equation help (:

The motion of a body falling in a viscous medium may be described by the equation:

m(dv/dt) = -bv-mg

where m, g, b are positive constants. The initial conditions are v(0) = 0 and x(0) = 0.Determine the velocity v(t) and displacement x(t) for t < 0.

2. ## Re: The motion of a body falling in a viscous medium differential equation help (:

\displaystyle \begin{align*} m\,\frac{dv}{dt} &= -b\,v - m\,g \\ \frac{dv}{dt} &= -\frac{b}{m}\,v - g \\ \frac{dv}{dt} + \frac{b}{m}\,v &= -g \end{align*}

This is now first-order linear, so solve using the Integrating Factor method.

3. ## Re: The motion of a body falling in a viscous medium differential equation help (:

Ok so I can get to v(t)=(-gm/b)(1+e^(-bt/m)) (is this correct?)

However, I am unsure now as to solve for x(t).

Sorry I have not yet learnt how to use the proper math writing on here.

4. ## Re: The motion of a body falling in a viscous medium differential equation help (:

I get \displaystyle \begin{align*} v = -\frac{g\,m}{b} \left( 1 - e^{-\frac{b}{m}\,t} \right) \end{align*}

Surely you know that velocity is the derivative of displacement, or the other way, displacement is the antiderivative of velocity...