- #1

- 209

- 2

## Main Question or Discussion Point

I have stuck at a step during derivation.

Here it is.

D=bv

mg-bv=ma

a=g-[tex]\frac{bv}{m}[/tex]

[tex]\frac{dv}{dt}[/tex]=g-[tex]\frac{bv}{m}[/tex]

[tex]\frac{dv}{g-[tex]\frac{bv}{m}[/tex]}[/tex] = dt

or,

[tex]\frac{dv}{\frac{mg-bv}{m}}[/tex] = dt

[tex]\frac{dv}{mg-bv}[/tex] = [tex]\frac{dt}{m}[/tex]

let, u=mg-bv

[tex]\frac{du}{dv}[/tex]=0-b

[tex]\frac{du}{dv}[/tex]=-b

multiplying both sides by, "-b"

[tex]\frac{-bdv}{mg-bv}[/tex]=[tex]\frac{-bdt}{m}[/tex]

[tex]\frac{du}{u}[/tex]=[tex]\frac{-bdt}{m}[/tex]

integrating L.H.S. from 0 to u and Integrating R.H.S. from 0 to t

[tex]\int^{u}_{0}\frac{du}{u}[/tex]=[tex]\frac{-b}{m}[/tex][tex]\int^{t}_{0}dt[/tex]

ln(u)|[tex]^{u}_{0}[/tex]=[tex]\frac{-bt}{m}[/tex]

ln(u)=[tex]\frac{-bt}{m}[/tex]

ln(mg-bv)=[tex]\frac{-bt}{m}[/tex]

mg-bv=e[tex]^{\frac{-bt}{m}}[/tex]

Now, I am stuck

L.H.S. must be [tex]\frac{mg-bv}{mg}[/tex]

but I am unable to get mg in denominator.

Where I am wrong?

Here it is.

D=bv

mg-bv=ma

a=g-[tex]\frac{bv}{m}[/tex]

[tex]\frac{dv}{dt}[/tex]=g-[tex]\frac{bv}{m}[/tex]

[tex]\frac{dv}{g-[tex]\frac{bv}{m}[/tex]}[/tex] = dt

or,

[tex]\frac{dv}{\frac{mg-bv}{m}}[/tex] = dt

[tex]\frac{dv}{mg-bv}[/tex] = [tex]\frac{dt}{m}[/tex]

let, u=mg-bv

[tex]\frac{du}{dv}[/tex]=0-b

[tex]\frac{du}{dv}[/tex]=-b

multiplying both sides by, "-b"

[tex]\frac{-bdv}{mg-bv}[/tex]=[tex]\frac{-bdt}{m}[/tex]

[tex]\frac{du}{u}[/tex]=[tex]\frac{-bdt}{m}[/tex]

integrating L.H.S. from 0 to u and Integrating R.H.S. from 0 to t

[tex]\int^{u}_{0}\frac{du}{u}[/tex]=[tex]\frac{-b}{m}[/tex][tex]\int^{t}_{0}dt[/tex]

ln(u)|[tex]^{u}_{0}[/tex]=[tex]\frac{-bt}{m}[/tex]

ln(u)=[tex]\frac{-bt}{m}[/tex]

ln(mg-bv)=[tex]\frac{-bt}{m}[/tex]

mg-bv=e[tex]^{\frac{-bt}{m}}[/tex]

Now, I am stuck

L.H.S. must be [tex]\frac{mg-bv}{mg}[/tex]

but I am unable to get mg in denominator.

Where I am wrong?