Given: v(dv)/dx = 9.8 - 0.004v^2, v(0) = 0vdvdx=9.8−0.004v2,v(0)=0
Use the separation of variables method:
v/(9.8 - 0.004v^2) dv = dxv9.8−0.004v2dv=dx
Integrate both sides:
intv/(9.8 - 0.004v^2)dv = intdx∫v9.8−0.004v2dv=∫dx
Multiply by 1 in the form of (-250)/-250:−250−250:
int((-250)/-250)v/(9.8 - 0.004v^2) dv = intdx∫(−250−250)v9.8−0.004v2dv=∫dx
-125int(2v)/(v^2-2450)dv = intdx−125∫2vv2−2450dv=∫dx
Let u = v^2u=v2 then du = 2dvdu=2dv:
-125int1/(u-2450)du = intdx−125∫1u−2450du=∫dx
ln(u-2450) = x/-125 + Cln(u−2450)=x−125+C
Reverse the substitution:
ln(v^2-2450) = x/-125 + Cln(v2−2450)=x−125+C
v^2-2450 = e^(-0.008x + C)v2−2450=e−0.008x+C
v^2= 2450 + Ce^(-0.008x)v2=2450+Ce−0.008x
Use the boundary condition:
0^2 = 2450+ C02=2450+C
C = -2450C=−2450
v^2= 2450 - 2450e^(-0.008x)v2=2450−2450e−0.008x
v^2= 2450(1 - e^(-0.008x))v2=2450(1−e−0.008x) Q.E.D.