How Does Constant Power Affect Rocket Acceleration in Space?

[DaveC426913]

In fact, leaving aside certain complications, it's the available power that is the constant. As speed increases, the same power yields a smaller accelerating force. Even without any resistance forces, acceleration will decrease. And, eventually, the accelerating force reduces to the point where even a small resistance force will equal it.

This seems wrong. I am thinking about a rocket in space. Ignoring the change in mass due to fuel loss, surely a rocket accelerates at a constant rate for a constant appliciation of force.

What am I missing?

(Have I forgotten to factor in the max velocity being limited to twice exhaust velocity?)

 

[PeroK]

This seems wrong.

It's not wrong, but a rocket is a special case. See below.

I am thinking about a rocket in space. Ignoring the change in mass due to fuel loss, surely a rocket accelerates at a constant rate for a constant appliciation of force.

Everything accelerates at a constant rate for a constant application of force.

What am I missing?

There are two ways to look at a rocket:

1) Assuming it is a closed system, then it cannot accelerate. What it can do is split apart and redefine a reducing fraction of the original system as the "rocket". There is no external force, as such. And the centre of mass of the original body of mass does not accelerate.

2) If we allow the rocket to be continually redefined, then it is capable of acceleration with the expellent providing an external force that is constant (even increasing) and there is not a constant power. It's continually redfining what is meant by "the rocket" that makes it a special case.

 

[DaveC426913]

It's not wrong, but a rocket is a special case. See below.

I chose a rocket because it is the simplest case. Surely, every other example is the special case.

Everything accelerates at a constant rate for a constant application of force.

OK, that is what I expected.

But it seems to fly directly in the face of what I quoted you saying.

2) If we allow the rocket to be continually redefined, then it is capable of acceleration with the expellent providing an external force that is constant (even increasing) and there is not a constant power. It's continually redfining what is meant by "the rocket" that makes it a special case.

You are talking about the rocket's decreasing mass. I specifically ignored that.

(Say my rocket is so big that its fuel loss for the test is miniscule by comparison, and so efficient that applies virtually all its energy as exhaust.)

Are you saying that such an ideal rocket - even in principle - will not maintain a constant acceleration for constant force?

 

[kuruman]

This seems wrong. I am thinking about a rocket in space. Ignoring the change in mass due to fuel loss, surely a rocket accelerates at a constant rate for a constant appliciation of force.

What am I missing?

(Have I forgotten to factor in the max velocity being limited to twice exhaust velocity?)

The rocket equation in the absence of external forces is $$m\frac{dv}{dt}=(u-v)\frac{dm}{dt}$$where
$m=~$ instantaneous mass of the rocket + unspent fuel
$v=~$ instantaneous velocity of the rocket relative to an inertial frame
$u=~$ instantaneous velocity of the exhaust gas relative to the same inertial frame

You can see that when $v=u$ the acceleration is zero.

 

[DaveC426913]

The rocket equation in the absence of external forces is $$m\frac{dv}{dt}=(u-v)\frac{dm}{dt}$$where
$m=~$ instantaneous mass of the rocket + unspent fuel
$v=~$ instantaneous velocity of the rocket relative to an inertial frame
$u=~$ instantaneous velocity of the exhaust gas relative to the same inertial frame

You can see that when $u=v$ the acceleration is zero.

OK, so you are confirming that the exhaust velocity is the limiting factor for rockets.

 

[kuruman]

OK, so you are confirming that the exhaust velocity is the limiting factor for rockets.

Yes. In the inertial frame where the rocket is initially at rest and no fuel has been spent, the total momentum is zero and must remain zero at all subsequent times. In that frame, the very first $dm$ of spent fuel has speed $u_0$ and the accelerating rocket cannot move faster than that in the opposite direction.

 

[PeroK]

I chose a rocket because it is the simplest case. Surely, every other example is the special case.

That doesn't make a lot of sense. Running, bikes, cars, trains, aircraft, boats etc. can't all be the special case. Rocket propulsion is the special case in this context.

But it seems to fly directly in the face of what I quoted you saying.

No it doesn't. It's the difficulty in maintaining a constant force that is the problem. To maintain a constant force, you need an increasing power supply.

You are talking about the rocket's decreasing mass. I specifically ignored that.

You can't ignore it!

(Say my rocket is so big that its fuel loss for the test is miniscule by comparison, and so efficient that applies virtually all its energy as exhaust.)

The question is how to maintain a constant acceleration. If the rocket starts as N particles, then those N particles cannot accelerate purely by an internal engine. This is an important observation, that cannot be neglected.

The way to achieve "acceleration" is by ejecting a particle in one direction, redefining the remaining N-1 particles as the rocket, which has now accelerated.

That would also work on, say, a train on smooth tracks. You could throw mass out the back of the train and have makeshift rocket propulsion. This is the exception, because trains don't work like that. They work by the wheels pushing on the track, which provides the external force.

Are you saying that such an ideal rocket - even in principle - will not maintain a constant acceleration for constant force?

You've completely misunderstood what I've said.

 

[PeroK]

This seems wrong.

There are a lot of people who think that applied force remains constant as speed increases. They see applied force as the fundamental quantity that can be maintained as speed increases.

 

[jbriggs444]

This seems wrong. I am thinking about a rocket in space. Ignoring the change in mass due to fuel loss, surely a rocket accelerates at a constant rate for a constant application of force.

What am I missing?

(Have I forgotten to factor in the max velocity being limited to twice exhaust velocity?)

Oh dear. The conversation has apparently shifted away from the question of gear ratios to the matter of a failure to account for the energy in a rocket exhaust stream.

As a rocket goes faster and faster there is an apparent gain in efficiency. The rate at which kinetic energy is gained by a rocket is proportional to the rocket's pre-existing velocity. With no apparent limit on how high the efficiency (craft KE gained per unit of fuel expended) can get.

The resolution to that conundrum is to examine the KE in the exhaust stream. As the rocket goes faster and faster, more and more KE is being lost by slowing down the fuel and throwing it out the back as exhaust.

In a complete energy accounting, the result is a wash. The total efficiency (total KE gained per unit of fuel expended) is the same regardless of craft velocity.

Comparing a car to a rocket is a lot like comparing apples and oranges.

 

[DaveC426913]

Oh dear. The conversation has apparently shifted away from the question of gear ratios to the matter of a failure to account for the energy in a rocket exhaust stream.

...

Comparing a car to a rocket is a lot like comparing apples and oranges.

The conversation got more and more generalized as we ostensibly got to the crux of the phenomenon. There was much discussion about power formulae, etc.

I never thought we were talking about gear ratios. In fact, I thought the goal was to abstract away from specific mechanisms and their associated quirks, such as static friction and gear ratios.

(A car is a little like a rocket - in the special case where the empty rocket's mass is~10³kg and the propellant mass is ~10²⁴kg).

Just goes to show how there are as many viewpoints as there are viewers.

 

[Dale]

What am I missing?

The exhaust. Humans are much closer to cars than rockets.

 

[kuruman]

Shown on the right is an excerpt from a January 13, 1920 New York Times editorial lambasting Robert Goddard's for his suggestion about firing rockets in space. The author of this unsigned editorial seems to think that Goddard's scheme will not work because in vacuum there is nothing to push against.

For those interested, here is the link to the full editorial:
https://graphics8.nytimes.com/packages/pdf/arts/1920editorial-full.pdf

 

[Dale]

Have I forgotten to factor in the max velocity being limited to twice exhaust velocity?

The max velocity is not limited to any value less than $c$.

You can see that when v=u the acceleration is zero.

But $v-u$ would never be zero for a functioning engine. It would only be zero if you gently drop the fuel off the rocket.

 

[kuruman]

But $v-u$ would never be zero for a functioning engine. It would only be zero if you gently drop the fuel off the rocket.

Yes, of course. I was thinking of something else. It's late where I am. Good night.