A 96 lb weight is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the speed. Find its velocity as a function of time if its terminal velocity is -128 ft/s.

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idealist10 · Answer

@SithsAndGiggles

anonymous · Answer

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The total force acting on the object is the sum of the resisting forces upward and the downward gravitational force.
$$m\frac{dv}{dt}=mg-kv$$
Solve this ODE for $v$ (a function of $t$). The terminal velocity is the velocity attained as $t\to\infty$, but in this case you want to be able to use it as a sort of initial condition. At terminal velocity, acceleration is zero; this is because the velocity attains a constant maximum. So, the terminal velocity must occur when $a=\dfrac{dv}{dt}=0$, i.e. when $mg-kv=0$, or $v=\dfrac{mg}{k}$.