# Please explain the physics of a spacecraft launching from the Moon or from Mars

**URL:** <https://boards.straightdope.com/t/please-explain-the-physics-of-a-spacecraft-launching-from-the-moon-or-from-mars/919189>\
**Category:** Factual Questions\
**Created:** [August 27, 2020, 10:23pm UTC](https://boards.straightdope.com/t/please-explain-the-physics-of-a-spacecraft-launching-from-the-moon-or-from-mars/919189 "2020-08-27T22:23:35Z")\
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**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\
**Post date:** [December 29, 2020, 11:24pm UTC](https://boards.straightdope.com/t/please-explain-the-physics-of-a-spacecraft-launching-from-the-moon-or-from-mars/919189/67 "2020-12-29T23:24:19Z")

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> [@md-2000](#):
>
> The answer is - hmmm, I’ll have to drag out my math textbooks and review Calculus.

The link @Stranger_On_A_Train gave is fine, but here’s a slightly different derivation of the Tsiolkovsky rocket equation:  
Start with:  
p=propellant flow rate (i.e., kg/s)  
m0=initial mass  
m1=final mass  
Ve=propellant exit velocity  
F=engine force

Then:  
m(t) = m0 - p⋅t  
And (since force is mass flow times exit velocity):  
F=p⋅Ve

Since F=ma and a=F/m:  
a(t) = p⋅Ve / (m0 - p⋅t)

We can integrate to:  
v(t) = -Ve⋅ln(m0 - p⋅t) + C

The final time is (when the propellant is depleted):  
t1=(m0 - m1)/p

Evaluating t from 0 to (m0 - m1)/p gives:  
delta v = -Ve⋅ln(m0 - p⋅(m0 - m1)/p) - -Ve⋅ln(m0 - p⋅0) = -Ve⋅ln(m1) + Ve⋅ln(m0) = **Ve⋅ln(m0/m1)**

And that’s it. You can convert to a form using Isp by substituting Ve=Isp⋅g. Note that the flow rate drops out of the final formula.

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