differentiation to touchdown

[pilotpercy]

hi guys, having a bit of a mental block at the moment. im in orbit round the moon with a view to landing at brighton beach. im all lined up and should be passing its overhead at roughly 700m in half an orbit. im now doing my calculations for my burn. i know the thrust of the DGIV with my fuel load is 12.2ms^(-2) (no turbo) an my velocity is around 1640ms^(-1). what im trying to find is the distance from brighton beach i need to start my burn, which i have calculated to be 134.42 seconds.

my mind has gone blank and i know i need to differentiate something as the ship will be slowing at a constant rate of 12.2ms(^-2) . can anyone help me out here?

cheers

 

[ar81]

If you are 300km above surface, you may start your deorbit burn when you are 300 km away (horizontally). Check the impact point. That's what I do.

 

[cr1]

Yea, since the moon has no atmosphere, the predicted impact point should be fairly accurate. Just make sure that when you get there you slow down enough, because if the periapsis is near the base, then probably the landing/impact speed is going to be quite high.

EDIT: Oh and sorry, I'm not so experienced with equations yet

 

[pilotpercy]

im in a 80k orbit but ive burnt retrograde so now the lowest point of my orbit is 700m over brighton beach. i know it will take 134.42 seconds for me to get my speed to 0ms^(-1) wrt brighton beach. im just trying to find the equation and the differential of said equation to give me the distance need to burn from. in a rough calculatinon i got the figure of 111.8km which worked quite well, as i stopped and went into the hover at around 400-300m from the landing pads but i want to refine my calculations.

basicly i need to differentiate speed with respect to time (i think).

 

[TJohns]

distance = (velocity squared) / (2 * acceleration)

d = (1640^2) / (2 * 12.2) = 110,229 m = 110 km plus change

or,

start at 1640 m/s, end at 0 m/s, average speed 820 m/s

820 m/s for 134.42 s travels 110,224 m

Trevor

 

[RisingFury]

You'll have to engage the hover thrusters... just burning the main engine horizontally will result in a nasty crash as the lowest point of your orbit goes below the surface.

I usually enter a very low orbit, a few km high, then at around 300 km from the base, I turn retrograde and burn for a bit, to lose some speed, still flying roughly 1 km/s. I engage my hover thrusters and start correcting my path towards the base. Then I wait until I'm a bit closer, about 100 km... then I lose more speed, lower it down to 300 m/s or so. All this time I keep on adjusting my hover thrust. I use the SurfaceMFD (I think it's called that) to cancel out my vertical acceleration. More course corrections follow. After that, I turn forward and start using my retro thrusters to lose velocity and I also switch RCS translation for some control. Soon I have Brighton in sight and velocity down low enough for nice approach. Final 100 meters are done with a very slow approach ~10 m/s and breaking although I have sometimes gone in WAY too fast

Then a smooth touch down.

 

[martins]

basicly i need to differentiate speed with respect to time (i think).

It is integration rather than differentiation you are after:

Integration of the equations of motion:
v(t) = v_0 + \int_0^t a(t) dt
x(t) = x_0 + \int_0^t v(t) dt

In your case (with constant acceleration a(t) = -a:
v(t) = v_0 -at
x(t) = x_0 - \int_0^t (v_0-at) dt = x_0 + v_0 t - 1/2 a t^2

Setting x_0 = 0, and t_0 = v_0/a:
x(t_0) = v_0^2/a - 1/2 v_0^2/a = 1/2 v_0^2/a

which gives you Trevor's formula:

distance = (velocity squared) / (2 * acceleration)

(sorry if the latex notation is a bit unreadable).

 

[Juanelm]

I think it would be more acurrate to vew the problem utilizing energy, since you need to take into account that the ship is high over the ground, and when it lowers its altitude, its speed is going to raise.

Ship Energy = Kinetic Energy + Potential Energy = .5(ship mass)*(current speed)^2+(shipmass)*(moon's gravitational acceleration)*(current altitude)

That will give you the total mechanical energy of the ship, which you need to bring to 0. To do this, you need to perform the same amount of work as the ship's energy, W=F*d. So, if you divide (Ship Energy)/(Retro Thrust Force) you should get the distance from the base you should start your burn. However, this is the trajectory distance, which must be approximated.

(did I get this all wrong...?)

 

[Scarecrow]

I think it would be more acurrate to vew the problem utilizing energy, since you need to take into account that the ship is high over the ground, and when it lowers its altitude, its speed is going to raise.

Ship Energy = Kinetic Energy + Potential Energy = .5(ship mass)*(current speed)^2+(shipmass)*(moon's gravitational acceleration)*(current altitude)

That will give you the total mechanical energy of the ship, which you need to bring to 0. To do this, you need to perform the same amount of work as the ship's energy, W=F*d. So, if you divide (Ship Energy)/(Retro Thrust Force) you should get the distance from the base you should start your burn. However, this is the trajectory distance, which must be approximated.

(did I get this all wrong...?)

Well presumably he'll be using his hover engines to slow his rate of descent, which will take care of the potential energy. The kinetic energy (in the horizontal direction) is all he needs to worry about for deceleration with his main engines.

 

[ar81]

What I do is to adjust the impact point when I am 300 km away from base at altitude of 300 km.
Then I do not let vertical speed go above 1km/sec, while in retrograde.
As I reach 15 to 30 km altitude I go leveled and use hovers.

A solution that is not worthy of a rocket scientist, but it works for me.

If engines are less powerful, vertical speed limit must be lower and it may take longer to brake.
I learned to pilot with WWII fighters so I guess I am handling the craft like an analogic equipped craft.

 

[cjp]

A solution that is not worthy of a rocket scientist, but it works for me.

One of the things you can do in Orbiter but not in real life is to simply try something and try again when it fails. For instance, try starting at 300km distance; when that makes you land 96km before the base, your next attempt will be 300-96=204km distance.

After one or two attempts you won't be able to make it more accurate, even when using calculations. In my case, I usually end up about 15km away from the base (before, after, left or right). Then, just hover, point your nose towards the base, and hover towards the landing pad.

I made a tutorial about this topic here.

 

[Arrowstar]

One of the things you can do in Orbiter but not in real life is to simply try something and try again when it fails. For instance, try starting at 300km distance; when that makes you land 96km before the base, your next attempt will be 300-96=204km distance.

After one or two attempts you won't be able to make it more accurate, even when using calculations. In my case, I usually end up about 15km away from the base (before, after, left or right). Then, just hover, point your nose towards the base, and hover towards the landing pad.

I made a tutorial about this topic here.

I do believe what you've described is essentially fixed point iteration of a solution. Granted, it's iteration by Orbiter flights and not mathematics, but it works the same.

 

[EtherDragon]

distance = (velocity squared) / (2 * acceleration)

d = (1640^2) / (2 * 12.2) = 110,229 m = 110 km plus change

or,

start at 1640 m/s, end at 0 m/s, average speed 820 m/s

820 m/s for 134.42 s travels 110,224 m

Trevor

You could use a similar formula for the Hovers - to determine when you should begin burning them (at full throttle) to bring yourself to a verticle speed of 0, at 0m. Just don't forget to take into account the y- acceleration of the Moon's Gravity.

[Edit]
It's a perfectly reasonable thing to use Math to cut out the guess work on this - If you reduce the time you use to decelerate to zero (by eliminating inefficient guessing) you reduce the fuel used to keep your ship above ground with the Hovers.

[Edit, edit]
Now the real trick would be the integration needed to account for your change in MASS over that time.
Mass(t) = StartingMass - FuelPerSecond*t
first integration yields your z- acceleration at t.
second integration yields your distance at t.