 # Given the equation V_m= u(ln m_0-ln m_8)- g t_f I need to solve for m_0 Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way Haiphongum 2022-09-18 Answered
Homework help to rearrange formula
Given the equation
${V}_{m}=u\left(\mathrm{ln}{m}_{0}-\mathrm{ln}{m}_{8}\right)-g{t}_{f}$
I need to solve for ${m}_{0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way
1st attempt
$\begin{array}{rl}& {V}_{m}=u\left(\mathrm{ln}{m}_{0}-\mathrm{ln}{m}_{8}\right)-g{t}_{f}\\ & {V}_{m}=u\left(\mathrm{ln}{m}_{0}\right)-u\left(\mathrm{ln}{m}_{8}\right)-g{t}_{f}\\ & {V}_{m}=u\left(\mathrm{ln}{m}_{0}\right)-u\left(\mathrm{ln}{m}_{8}\right)-g{t}_{f}\\ & {V}_{m}=u\left(\mathrm{ln}{m}_{0}\right)-u\left(\mathrm{ln}{m}_{8}\right)-g{t}_{f}\\ & u\left(\mathrm{ln}{m}_{0}\right)-u\left(\mathrm{ln}{m}_{8}\right)-g{t}_{f}-{V}_{m}=0\\ & u\left(\mathrm{ln}{m}_{0}\right)-u\left(\mathrm{ln}{m}_{8}\right)-{V}_{m}=g{t}_{f}\\ & u\left(\mathrm{ln}{m}_{0}\right)=g{t}_{f}+u\left(\mathrm{ln}{m}_{8}\right)+{V}_{m}\\ & \mathrm{ln}{m}_{0}=\left(g{t}_{f}+u\left(\mathrm{ln}{m}_{8}\right)+{V}_{m}\right)÷u\\ & {e}^{\left(g{t}_{f}+u\left(\mathrm{ln}{m}_{8}\right)+{V}_{m}\right)÷u}={m}_{0}\end{array}$
2nd attempt - think this looks a little better but still not there yet
$\begin{array}{rl}& {V}_{m}=u\left(\mathrm{ln}\frac{{m}_{0}}{{m}_{8}}\right)-g{t}_{f}\\ & {V}_{m}+g{t}_{f}=u\left(\mathrm{ln}\frac{{m}_{0}}{{m}_{8}}\right)\\ & \frac{{V}_{m}+g{t}_{f}}{u}=\mathrm{ln}\frac{{m}_{0}}{{m}_{8}}\\ & {e}^{\frac{{V}_{m}+g{t}_{f}}{u}}=\frac{{m}_{0}}{{m}_{8}}\\ & {m}_{8}{e}^{\frac{{V}_{m}+g{t}_{f}}{u}}={m}_{0}\end{array}$
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Hint:
$\mathrm{ln}\left(a\right)-\mathrm{ln}\left(b\right)=\mathrm{ln}\left(\frac{a}{b}\right)$

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