Find the displacement and velocity of horizontal motion in a medium in which the retarding force is proportional to the velocity.
I kind of understand how to do this problem.
We know that the resistive force $F_r \propto v$. Since $F_r$ is the only force present in the x-direction, Newton's second law gives $$F_r=ma=m\frac{dv}{dt}.$$ My book then says that $F_r=-kmv$. So thus we have $$-kv=\frac{dv}{dt},$$ from which it is trivial to find expressions for $v(t)$ and $x(t)$ by using initial conditions and integration.
The only part about the problem I don't understand is why $F_r=-kmv$. Why does the retarding force depend on the mass $m$? Since $F_r \propto v$, shouldn't we just stick a proportionality constant $k$ in there and have $F_r=-kv$?
Best Answer
Your physical intuition is correct. A resistive force arising from motion in a viscous medium should not depend on the mass of the object. See, for example, Stokes drag for a common model of this kind of resistive force. So it is likely that the force is defined this way to make the equation of motion look nice. If you used a different object with a different mass, $k$ would have to change accordingly.