Derivation of a variant of the Tsiolkovsky rocket equation which includes gravity

Take the rocket equation :

v = ln(m0/mf) ve - g tburn

Where tburn is time to burn, v is velocity,  ve is exhaust velocity,  m0 is initial mass , mf is final mass

The Apollo third stage started its run to escape velocity at an altitude of 191km and speed of 8km/s. Per NASA.

Set Ve equal to the Saturn IVB exhaust velocity, or 4km/s. The fuel mass is 104 tons. Dry mass is 50t with payload. 475 sec burn time. G is 9.2m/s2 at 191km altitude.


Dv =  ln(104/50t) 4kms - 9.2m/s2 (475s)

First term is

= 4kms (0.73)

= 3kms

Second term is

  • 9.2m/s2 (475s)

= -4370 m/s


3kms - 4.37 kms = -1.37 km/s

The Saturn IVB would have a maximum velocity of -1.37 km/s which is far less than the earth escape velocity of 11km/s, and is still less even if we give it a 8km/s head start.

  • $\begingroup$ There is no actual question, but you might need to consider that the SIVB was not accelerating straight upwards. $\endgroup$ – Steve Linton Jun 9 '18 at 23:38
  • $\begingroup$ Voted to close as there doesn’t appear to be a question here. As for the calculation - the g term is not included because the trans-lunar injection impulse is perpendicular to the gravitational acceleration (approximately) $\endgroup$ – Jack Jun 9 '18 at 23:45
  • $\begingroup$ There are plenty of good resources on this site and elsewhere that can help you understand how to calculate orbital manoeuvres. If you still can’t find what you’re looking for, please read the Help Center, then ask a new question. $\endgroup$ – Jack Jun 10 '18 at 0:09
  • $\begingroup$ You should use the LAST formula in this answer space.stackexchange.com/questions/15509/… $\endgroup$ – Heopps Jun 10 '18 at 6:49
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    $\begingroup$ Possible duplicate of Saturn S-IVB math $\endgroup$ – user20636 Jun 10 '18 at 8:06

3kms - 4.37 kms = -1.37 km/s

This is yet another case of what goes wrong when you try to treat delta-v as a scalar.

As @Jack points out, these accelerations are not co-linear, so you can not add them algebraically. You need to treat acceleration as a vector, and consider the direction that the thrust is accelerating and the direction that gravity is pulling.

Any satellite in a stable orbit also experiences constant gravitational acceleration for years or decades, and yet it doesn't fall out of the sky.

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    $\begingroup$ Or rather, it keeps trying to fall out the sky, but it just has terrible aim ;) $\endgroup$ – Jack Jun 9 '18 at 23:56
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    $\begingroup$ delta-v is a scalar. how you apply it is a different matter. $\endgroup$ – user20636 Jun 10 '18 at 8:16
  • $\begingroup$ @JCRM Indeed. it was only a matter of time before I talked about delta-v as a scalar myself. space.stackexchange.com/q/28026/12102 $\endgroup$ – uhoh Jun 22 '18 at 9:01

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