-1

Why the following would not work?

A manager of a deep space station has 500 hours to deliver a 10,000 tons cargo ship to a neighbor 1,000,000 miles away. For this to happen the ship has to travel at average speed of 2,000 miles/hour. The ship engine converts fuel's energy into ship's kinetic energy and it needs 1 ton of fuel to accelerate to 1,000 miles/hour. To reach 2,000 miles/hour the engine needs 4 tons of fuel (4x kinetic energy at 2,000 compare to 1,000 miles/hour).

The manager has 2.1 tons of the fuel, and his solution is:

  1. Load ship with 2 tons of fuel.
  2. Turn off the engine after the ship used 1 ton of fuel and reached speed of 1,000 miles/hour.
  3. On the manager's own tiny capsule he matches the cargo speed, then turn on the cargo ship engine again (remotely).
  4. After burning another 1 ton of the fuel, ship's speed will be 1,000 miles/hour in the capsule's reference frame and 2,000 relative to the space stations.
  5. Total fuel use was 2 tons, instead of 4.
Arcady Grenader
  • 9

1 Answers1

1

The ship engine converts fuel's energy into ship's kinetic energy and it needs 1 ton of fuel to accelerate to 1,000 miles/hour. To reach 2,000 miles/hour the engine needs 4 tons of fuel (4x kinetic energy at 2,000 compare to 1,000 miles/hour).

This is how cars work. By using the mass of the earth to push against, they can put nearly all the energy of fuel into kinetic energy of the car (ignoring drag).

But rockets don't work this way. Most of the energy from the fuel goes into kinetic energy of the exhaust. Only a small fraction goes into the energy of the vehicle. But as the burn goes on, the fraction goes up. So double the speed of the vehicle does not require double the fuel use.

The manager doesn't need to accelerate the capsule. The initial premise about needing 4x the fuel for double the speed is incorrect (for regular rockets). It might be possible if the ship isn't a rocket (maybe it's accelerating against a planet's powerful magnetic field). But that changes lots of things. Mainly what the speed of the craft is.

For a car, the ability to accelerate goes down as the thing you're pushing against (the ground) is moving faster. If your ship is pushing against something, then that's the speed that matters. The speed relative to the capsule is irrelevant.

Double speed means 4x kinetic energy. If all ship's kinetic energy comes from the fuel, 4x kinetic energy -> 4x fuel

There are at least 2 incorrect assumptions in that statement.

The first is that all the energy is going into the vehicle KE. That's not true. Some (actually, a lot) is going into the KE of the exhaust. Imagine this for how things are working:

  • First second: 1000J fuel -> 1J KE rocket, 999J KE exhaust
  • Second second: 1000J fuel -> 4J KE rocket, 996J KE exhaust

The second assumption is that the fuel has the same energy all the time. It has the same chemical energy all the time, but later on (same as "in a frame where the rocket has high forward velocity") it also has some KE. And it can give up that KE to give additional energy to the rocket.

The result of both is that as long as the rocket mass remains relatively constant, then the same fuel burn gives the same velocity change. In reality as the rocket loses mass, the same fuel burn will give more velocity (even though the vehicle is getting a much greater amount of energy in that frame).

This is completely OK due to conservation of energy, but you have to account for the KE of the fuel and the KE of the exhaust as well.

BowlOfRed
  • 40,008
  • "The initial premise about needing 4x the fuel for double the speed is incorrect" Double speed means 4x kinetic energy. If all ship's kinetic energy comes from the fuel, 4x kinetic energy -> 4x fuel (doesn't matter what fraction of the fuel chemical energy was converted into ship's kinetic energy) – Arcady Grenader Nov 29 '22 at 18:59
  • How is it relevant if 100% or 0.01% of the fuel was converted into ship's kinetic energy? – Arcady Grenader Nov 29 '22 at 19:06
  • Added a bit to the answer. – BowlOfRed Nov 29 '22 at 19:13
  • "And it can give up that KE to give additional energy to the rocket." It can matter if fuel mass is comparable or larger than the ship mass. The question isn't about real current rockets where most of the mass is fuel. It's a thought experiment. – Arcady Grenader Nov 29 '22 at 19:20
  • All that matters is how the rocket moves. If it is exhausting onboard fuel backward at high speed, then X fuel burn ~ Y deltaV and you get approximately constant thrust/acceleration. If it is pushing on something external (like a planet or a fixed magnetic field), then you get constant power and need the square of the fuel to go double the speed. – BowlOfRed Nov 29 '22 at 19:41
  • Let's only deal with rocket in the deep space. The only source of the rocket kinetic energy is its fuel, where some fraction of fuel chemical energy is converted into ship's KE. If this fraction doesn't depend in the ship's speed, the 4x KE would require 4x fuel. What I'm missing? – Arcady Grenader Nov 29 '22 at 20:55
  • https://physics.stackexchange.com/questions/428952/why-is-the-work-done-by-a-rocket-engine-greater-at-higher-speeds, https://physics.stackexchange.com/questions/44884/how-can-kinetic-energy-be-proportional-to-the-square-of-velocity-when-velocity, and https://physics.stackexchange.com/a/341815/55662 are all relevant – BowlOfRed Nov 29 '22 at 21:15
  • Thanks! I'll take a look. – Arcady Grenader Nov 30 '22 at 00:34
  • Thanks a lot BowlOfRed! The references really helped. I forgot that we have to look at the ship's KE + exhaust KE. The faster ship is moving, the higher fraction of the fuel's energy is converted to the ship's KE at the expense of the fuel's KE. – Arcady Grenader Nov 30 '22 at 05:18