I need to calculate the minimum time, t, it takes to travel a given distance, x.
Constraints and Known Variables
Distance x is broken up into two distinct, known segments:
- xv = vacuum distance
- xa = atmospheric distance
- x = xv + xa
While traveling the length of distance xv, I am under constant, known acceleration or deceleration:
- ava = vacuum acceleration rate
- avd = vacuum deceleration rate
While traveling the length of distance xa, I am under constant, known acceleration or deceleration:
- aaa = atmospheric acceleration rate
- aad = atmospheric deceleration rate
I will have known velocity constraints:
- v0 = starting velocity
- vf = ending velocity
- vmax = max velocity (cannot exceed this velocity)
I'm in a spacecraft 60km above the surface of a moon. I need to travel to the surface of the moon as quickly as possible. Ignore rotations/orbits. Both my starting and ending velocities are/must be 0 m/s. The moon's atmosphere begins sharply at 30km (there is no gradient here). I have four distinct acceleration/deceleration rates: acceleration in a vacuum, deceleration in a vacuum, acceleration in atmosphere, and deceleration in atmosphere. My spacecraft is governed by a top speed. What is the shortest amount of time it takes me to reach the surface.
I'm in a spacecraft on the surface of the same moon. I need to climb to an altitude of 45km as quickly as possible (remember, the atmosphere ends sharply at 30km). My starting velocity is 0 m/s, but I do not care what my ending velocity is, so long as it does not exceed the top speed of my craft.
I've solved the above problem for a single rate of acceleration and a single rate of deceleration. However, the addition of the notion of two distinct rates of acceleration and two distinct rates of deceleration has really thrown me through a loop. I'm pretty much lost after banging my head against a wall for a few days.