# Problem

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)

# Example 1

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.

# Example 2

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.

# Question

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.

• why not solve for the two situations separately? it seems they can be thought of as distinct problems. – The Count Aug 29 '18 at 2:16
• I second the above comment. You have starting velocity. First, find velocity at the point were atmosphere ends and that's your initial velocity for the second part of the problem. – Vasya Aug 29 '18 at 2:26
• They are not entirely distinct problems. If my ending velocity was 0 and my final velocity coming out of segment 1 and into segment 2 was too large to decelerate from, I've got a problem. I'd have to cap my segment 1 final velocity to the max allowable velocity entering segment 2, considering ending velocity requirements and rate of deceleration. They are inter-related insofar as segment 1 must respect the overall problem's constraints with respect to what would happen in segment 2. – Daniel A. Burke Aug 29 '18 at 16:43
• Well, it seems you've since gotten it sorted out. At a glance, i would solve this using letters in place of the relevant starting and ending velocities, and then use those results to bound the relevant things. – The Count Aug 30 '18 at 1:25

For the second, accelerate fully in the atmosphere until either you hit top speed or you hit the top of the atmosphere. In the first case, just keep going to $45$ km. In the second, keep accelerating in vacuum until maximum speed, then just hold it.