Help Calculating Rocket Flight Height in Kerbin

[Driger]

Hi everyone,  I’m trying to predict the maximum height of my rocket flying straight up in the kerbin atmosphere and am stuck.

My ship is one flea booster with a Mk1 pod and a Mk16 chute flown by a kerbal. It weighs 2.44t full and 1.39t empty. The flea has an ISP of 140 asl - 165vac. Using the asl ISP I got 772.0166 delta v. From here I used some kinematics equations and figured that the speed at the end of the burn should be 685.16m/s and the maximum height of the flight would be 23926.72m

 When I tested it out in the sandbox with drag turned off I ended up going 29606m high and the speed at the end of the burn was 704m/s

So I think the numbers were off because as the rocket flys up the ISP improves meaning I actually had more delta v than calculated. Another thing I thought might have affected it was the force of gravity changing the further away from kerbin I got (I used a constant 9.81).

How do I improve my calculations to account for changing ISP and gravity? Or was I missing something else?
If anyone is feeling ambitious I would also really appreciate an explanation of how to add drag in as well because the formulas on the atmosphere wiki are not making any sense to me.
 

[Venusgate]

I'd say discount "gravity weakens as you go higher", as you maybe went from 9.81 m/s^2 to 9.805 in that distance.

Other factors to consider:
atmospheric drag is going to take some dV off.
You are already going ~250m/s tangentially eastward when you are landed, so calculating surface velocity can be kinda wonky.

[Snark]

25 minutes ago, Venusgate said:

I'd say discount "gravity weakens as you go higher", as you maybe went from 9.81 m/s^2 to 9.805 in that distance.

Actually, gravity does change significantly.  Bear in mind that these are toy planets with radii that are much smaller than IRL.

Gravity on Kerbin at an altitude of 24 km is 92% of sea level, or about 9.07 m/s2.

For the OP:  You're going to have difficulty coming up with exact numbers, since the Isp is changing, and the burn is proceeding over time (rather than an instantaneous impulse).  If you add in drag, it gets very hard (read: essentially impossible) to have an analytic solution; you pretty much have to do numerical integration.