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Daquicker

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## Homework Statement

I'm trying to calculate the absolute lowest safe altitude above a planet's surface to start firing the thrusters on a simulated lander in order tot just reach vCraft = 0 at the planet's surface.

The force the craft's thrusters responsible for desceleration generate is however dependent on the density of the atmosphere they're in, which in turn depends on the altitude above seaLevel they're currently at.

I'm looking for a general solution that returns an altitude H.

__general parameters of the problem:__

- Drag is not to be taken into account.
- Craft mass does not vary during descent. (Electricity fuels the thrusters so no solid/liquid fuel being burnt).
- Craft falls straight down along the gravity vector of the planet.
- Thrusters providing desceleration are pointed straight down along the gravity vector.
- The planet does not rotate.
- Planet g does not change. (At applicable altitudes change will be negligible).

__Data accesible from telemetry:__

general:

general:

- CraftAltitude above surface: xSurf is known at any time (in m)

- CraftAltitude above seaLevel: xSea is known at any time (in m)

- CraftVelocity (downwards): vCraft is known at any time (in m/s)

- CraftMass: mCraft is known at any time (in Kg)

- CraftMaxThrust at SeaLevel: xMaxThrustSea is known at any time (in Newton)
- Planet g: is known at any time (in m/s^2)

__for this example:__

- Atmosphere density: Altitude = 0m -> AtmDens = 1 | Altitude = 14400m -> AtmDens = 0
- Thruster efficiency: AtmDens = 1 -> ThrustEff = 1 | AtmDens = 0.3 -> ThrustEff = 0

## Homework Equations

- Work = m * g * d | m: mCraft, g: aCraft, d: xSurf

- Kinetic Energy = (m * v^2) / 2 | m: mCraft, v: vCraft

## The Attempt at a Solution

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I started with the following simple and common idea:

Get an altitude from the Work required to dissipate the craft's Kinetic Energy

=> W = KE => d = (v^2) / (2 * g)

This however, fails to account for the change in g due to the change in the craft's altitude above sea level (xSea), instead always using the theoretical maximum thrust. The consequence being it gravely underestimates the required braking distance.

I then came up with the following function which describes the craft's instantanious thrust at an altitude xSea:

CraftMaxThrust(xSea) = cMaxThrustSea - (cMaxThrustSea / 100) * ((100 / 0.7) * (1 - (1 - xSea / 14400)))

The acceleration at an altitude xSea provided by the thrusters would then become:

a(xSea) = CraftMaxThrust(xSea) / mCraft

So I figured I needed some type of differential equation to adjust for this shift in force.

This is where I get stuck, all I can come up with is that it should probably be something amongst the lines of:

0 = vCraft + ∫(H->0) |some derivative of a(xSea)?| a(xSea)

With vCraft being initial velocity, a(xSea) adjusted with the planet's g, and H the altitude I've been looking for all along.Eventhough it's not an actual homework assignment, any help would be greatly appreciated, it's been evading me for a couple of days now.

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