B Does Gravity decrease at a steady rate as we go away from the Earth?

[Dale]

launch efficiency is greatest for a given delta-v when you burn your fuel fast -- giving gravity less time to reduce your velocity. [In the limit you get zero efficiency if you sit there hovering on your engines].

This is also why they tip over as soon as they can get out of the atmosphere. The important thing is to get to the right speed, and that works better the more you go sideways and the less you go up.

 

[hutchphd]

It turns out that the human capsule requires a much less pronounced "tip" on the way up because of possibility of an abort. Too much altitude with no forward speed apparently produces an unfortunate reentry profile involving high g loads for a ballistic reentry. So the crewed rockets don't go straight but ascend at an increasing angle.
I grew up watching project Mercury et seq and I am a true space geek.

 

[mfb]

Space X rockets have no fuel pump. They use 300 psi helium to pressurize the liquid methane & liquid oxygen. SpaceX engines run about 1/4 throttle compared to German technology.

I believe they were talking about the new 8 meter diameter rocket bodies. The older rocket bodies were 5 meters diameter.

This is a strange combination of different rocket engines and rockets.
SpaceX developed three rocket engines, two of them are still in operation:

• Kestrel: Retired, was used in the upper stage of Falcon 1. Pressure-fed with liquid oxygen and RP-1 (basically kerosene) at 9 bar (135 psi). Thrust was 30 kN.
• Merlin: Active, used in both stages of Falcon 9. It uses fuel pumps. Burning liquid oxygen and RP-1 at 97 bar (1410 psi). Thrust increased from 350 kN to 850 kN over its multiple iterations.
• Raptor: Active, used on Starship prototypes. It uses fuel pumps. Burning liquid oxygen and methane at up to ~300 bar (4400 psi). Thrust is about 2000 kN.

Falcon 1 had a diameter of 1.7 m, Falcon 9 has a diameter of 3.7 m, Starship has a diameter of 9 m.

I am trying to understand when fuel runs out how far can a rocket coast with no engine thrust before it reaches maximum distance from earth.

Rocket engines are irrelevant for that. You only need the velocity and the height when the thrust stops. How the rocket got there doesn't matter.

Typically suborbital flights don't go higher than 600 km or 1/10 the radius of Earth. At that altitude you still have 80% the gravitational acceleration you have on the ground. For a rough estimate you can neglect that change and simply use the constant acceleration equations you learn in high school.

In 1930 it is interesting Germany knew with no knowledge of rockets that a rocket would coast 70 more miles up after engine was off. The law of motion probably let then calculate distance assuming they knew the value of gravity at 120 miles up.

Of course they knew. They even had experimental data from 400,000 km up - from the Moon.

This is also why they tip over as soon as they can get out of the atmosphere. The important thing is to get to the right speed, and that works better the more you go sideways and the less you go up.

It's not very notable, but typically rockets start that as soon as they clear the launch pad. Thrust goes in the (now slightly tilted) direction of the rocket, gravity goes downwards, so over time the trajectory gets flatter on its own while the rocket always flies and fires along its length to minimize aerodynamic forces.

 

[mpresic3]

I have an idea that a rocket engine can gradually throttle down at the same rate that gravity degreases and maintain the same speed because at a certain elevation there is no atmosphere and no wind resistance.

This is an interesting idea, but what would this buy you. Why would you want to throttle down to maintain constant speed. In your diagram of the V-2 flight (very informative, by the way), there is an event called "jet switched off at correct range velocity combination.

Historically (and presently), The rocket scientist / military targeteer calculates at all times on the trajectory path, (in practice this would be a calculation done onboard the rocket, with input from the inertial guidance system which would give the control system, the current rocket location):

1. Given the target location, the rocket current position, and time of flight to target
2. What velocity is needed for the rocket to free-fall to the target? (This is called correlated velocity)

After this velocity is calculated, the rocket control system directs the thrust, (by controlling the nozzles), to "speed" towards this " correlated" velocity.

The reason this procedure is used is because, scientist/mathematicians/engineers, have known for 100 years or so, how to calculate the correlated velocity from the current position, and time of flight. Control system engineers know how to direct the nozzles to continuously approach the correlated velocity. So thrust is terminated and free-fall begins at the correlated velocity.

Probably the most important aspect to all this is the problem in control theory. In my personal view, (this might not be shared by more informed engineers and historians), the Wright Bros success was due to their advances in control (ideas in wing warping, and others), more so than (for example) propulsion.

Thrust to maintain constant speed rather than thrust cut-off (called switch off), would involve harder calculations that calculating the correlated velocity, and what would be the advantage?

With the poster's background in Mechanical and Electrical Engineering, I would recommend books in Aero-engineering like Siouris, or Vallado, or Bate, Mueller, White (fundamental of Astrodynamics). I will be glad to add titles to these authors if requested. I do not have them at my fingertips, right now