Question

what is the limiting velocity of the object?

m = 100kg
g=9.8 m/sec^2
b=5kg/sec
v(0)=10m/sec
v(t) = ?

m*(dv/dt)=mg-bv

Answer 1

So you have a linear ODE:
\[100\frac{dv}{dt}=100\cdot9.8-5v(t)\\
\frac{dv}{dt}+\frac{1}{20}v(t)=980\]
Find the integrating factor:
\[\large\mu(t)=e^{\int\frac{1}{20}~dt}=e^{\frac{1}{20}t}\]
\[e^{1/20~t}\frac{dv}{dt}+\frac{1}{20}e^{1/20~t}v(t)=980e^{1/20~t}\]
Note that the left side is a derivative:
\[\frac{d}{dt}\bigg[e^{1/20~t}~v(t)\bigg]=980e^{1/20~t}\]
Integrate both sides:
\[e^{1/20~t}~v(t)=980\int e^{1/20~t}~dt\]
And so on. Once you solve for \(v(t)\), the limiting velocity will be given by \(\displaystyle \lim_{t\to\infty}v(t)\).