Calculate Altitude

[Brian W.]

Has anyone figured out a quick/easy way to calculate how high each stage of your craft will take you?  I thought I had stumbled on to delta-V*(burn time/2) as a plausible way to get a good approximation but in my testing in sandbox mode I found I am completely off when it comes to multistage setups.

Edited June 2, 2020 by Brian W.

[jimmymcgoochie]

There are a whole lot of variables that will determine how high your rocket will go- mass, fuel quantity, fuel efficiency, thrust, thrust to weight ratio, aerodynamic drag, launch profile and many more, not to mention what planet/moon you're on. Launching straight up will get more altitude from your stages, but you need to go sideways to reach orbit which gains velocity instead of altitude.

[Streetwind]

When flying straight up, you are fully subject to gravity. Every second, your craft is accelerated towards the gravity well by the value of gravity. For example, on Kerbin, every second you are hovering or moving straight up and down, you add 9.8 m/s velocity in the direction of the planet's core.

You need engines to overcome this. If your engine also adds 9.8 m/s of velocity to your craft every second, but in the direction away from the planet's core, your craft experiences no net change in velocity - you are hovering (or rather, holding your current velocity). This is referred to as a thrust-weight-ratio ("TWR") of 1.0. If your thrust-weight ratio is greater than 1, you accelerate away from the gravity well. If it is less, you are pulled down.

Gravity changes slowly with altitude. If you are gaining altitude, that means that gravity is decreasing, which means less velocity is added in the direction of the gravity well, which means that the same amount of engine power will accelerate you faster. Your effective TWR increases with altitude. Do note, however, that whenever people talk about TWR, they generally do so in regards to surface gravity, so don't get confused.

As your engine consumes fuel, the weight of your rocket decreases. This, again, improves your effective thrust-weight ratio, meaning that the same amount of engine power will accelerate you faster as you progress through your flight. Rockets accelerate at their hardest right before burnout.

Finally, if you are launching from Kerbin (or any other body with an atmosphere), the performance of your engine is diminished by atmospheric pressure. As you climb out of the atmosphere, this effect gradually disappears - the engine returns to its full power output. This again means that your effective TWR increases as you gain altitude. At the same time, the atmosphere also causes drag against the craft, which works to reduce its velocity. Atmospheric drag depends strongly on the shape of the object and its speed. As the atmosphere thins, this effect gradually disappears.

As you can see, there is a large amount of factors in play, many of which are unique to each individual vessel and/or launch location. The math you need to do is almost guaranteed to involve integrals of functions over the burn time and other such shenanigans. Considering that there's no practical gameplay reason to answer that question, it's probably not worth the effort - unless you really must satisfy your curiosity. In which case, good luck  

[Brian W.]

Thank you for the VERY DEEP insight, Street.  I was hoping to find a way, other than trial and error experimenting, to calculate reaching a certain altitude in order to complete some contracts.