Time to travel a given distance, given two separate, but constant acceleration and deceleration rates and a max velocity.

kinematicslinear algebraphysics

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.

Best Answer

You need to figure out where the breakpoints are, then it just becomes solving constant acceleration between the breakpoints. For your first example, you probably accelerate up to maximum speed, stay at that speed until you need to decelerate, and decelerate fully. The only reason this would not be true is if you can't get up to maximum speed before you need to decelerate, so take that assumption. Now see if you can get to maximum speed before the atmosphere starts. If you can, you just accelerate in vacuum up to maximum speed. See if you can decelerate from maximum speed while within the atmosphere. If you can, do that, and you have the breakpoints where you stop accelerating and start decelerating. If you can't decelerate within the atmosphere, decelerate fully in the atmosphere and solve for the velocity you need at the top of the atmosphere. Start decelerating in vacuum to meet this speed.

If you can't get up to max speed and back down again, you need to locate the altitude of maximum velocity and the velocity at that altitude. You will accelerate fully until you hit that point, then decelerate fully. Guess whether it is in or out of the atmosphere. One of the regimes will just have one acceleration rate, the other will have two.

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.

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