How do you calculate gravity loss (simplified)?

Assume a rocket with constant mass and thrust. To simplify things, we can ignore aerodynamic drag, and we'll also not perform a gravity turn, hence our flight plan is as follows: -Burn radially from sea level until apogee is 100km -Burn horizontally when apogee is 100km

How do you calculate the gravity loss from launch until apogee ( are there any equations)? If I remember correctly, it also depends on the TWR.

Secondary question: Does burning perfectly horizontally ever occur in any gravity loss? Is it best to burn horizontally straight away when apogee reaches 100km, or wait until apogee? (The Oberth effect states to burn when closest to the planet).

Dimensional analysis is a zeroth-order way to approximate (thank you Mr. Ross; 9th grade Physics). There will likely be better answers, but let's see what happens.

meters/sec^2 x seconds = meters/sec

9.8 m/sec^2 x 150 sec (MECO, where you're going mostly sideways) gives ~1500 m/s delta-v

People usually give something like 0.9 to 1.5 km/s when forced to cough up a number, so dimensional analysis works pretty good in this case.

• Thanks. How would you calculate the burn time/ ∆V required to get apoapsis to a certain height (burning straight up)?
– Sams
Jul 23, 2019 at 6:04
• @Sams that's a new question. In Stack Exchange we limit questions to question posts and answers to answer posts. Comments are really only for use in clarifying the posts. You can consider asking that as a new question but when you talk about "apoapsis" and "burning straight up" I think you will need to take some time and explain exactly what you mean.
– uhoh
Jul 23, 2019 at 6:06
• Alright, thanks. I'm new here, so don't know the rules.
– Sams
Jul 23, 2019 at 15:27