Why Does Kinetic Energy Increase So Much in a Constantly Accelerating Rocket?

[D H]

No, it isn't. You are insisting on ignoring the exhaust, Bob.

 

[Bob_for_short]

If an engine pulls the probe mass with a rope, then there is no exhaust. I am speaking of a general case.

 

[Dale]

By the way, to the OP, although it is certainly possible to analyze a rocket motion from the perspective of the conservation of energy it is too easy to neglect the energy of the exhaust as you and others have done. Therefore it is my recommendation to always work rocket motion problems from the perspective of the conservation of momentum. You will always get the same answer, but the conservation of momentum approach forces you to consider the exhaust.

 

[Bob_for_short]

Maybe but what if I want to consider a body of a constant mass? Why should I reduce the generality? The paradox of the problem is in neglecting the kinetic energy of the engine. It is the true source of the additional power. Power is a frame-dependent quantity.

 

[Dale]

Yes, Bob_for_short, I know, I am only talking about common mistakes made by beginning students when analyzing rockets and a simple way to avoid them. The OP's question is not about general engines, but specifically about rocket engines. I recommend that students use conservation of momentum for rocket problems whenever possible, simply because it avoids mistakes.

I am sure that you, Bob_for_short, will do fine and avoid mistakes using other approaches that will cause problems for students. So feel free to ignore my recommendation as it does not apply either to you nor to general engines.

 

[rcgldr]

If an engine pulls the probe mass with a rope, then there is no exhaust. I am speaking of a general case.

If there is a rope in involved then you have interaction with an external object, the point of appllication of force is upon the rope, and with constant power the rate of force will decrease with speed with respect to the rope according to the equation power = force x speed. This changes the situation from the rocket and its spent fuel, which is a closed internal interaction that increases kinetic energy of both rocket and fuel, but does not change the center of mass or the total momentum of the rocket and fuel system.

In the rocket case, the point of application is between the engine and the remaining on-board fuel. Regardless of the speed of the rocket and onboard fuel relative to some frame of reference, the engine and onboard fuel always have zero relative speed. For a given engine at a specific throttle setting the mass flow, thrust, power, and terminal velocity of the spent fuel relative to the rocket is constant, and the kinetic energy of the total system of rocket and spent fuel increases linearly with respect to time, regardless of the frame of reference.

 

[Bob_for_short]

Let me put it in a simple way; the power P is a frame- or velocity-dependent quantity:
P = Fv.

The force and acceleration are Galilean invariants but velocity is not. So in different RFs the powers are different.

Or in the same RF but at different time moments - when the body is moving slowly or quickly, the velocities are different, so are the powers.

 

[rcgldr]

Let me put it in a simple way; the power P is a frame- or velocity-dependent quantity: P = Fv.

Power can be calculated to avoid making it frame dependent. The rate of consumption of potential chemical energy of the fuel times an effeciency factor allows power to be calculated independent of the frame of reference. The work peformed (change in KE) per unit time on spent fuel and rocket is constant (for a given throttle setting) regardless of the frame of reference and would provide another means to calculate the power independent of the frame of reference.

Power is defined as a rate of work, from which the force x velocity form can be derived. In the case of a rocket, in order to use the force x velocity form, velocity should be defined based on the change in velocity of the fuel and rocket, or change in velocity of the fuel with respect to the rocket, which are frame independent.

DH's rocket tutorial thread covers this:

Power
The time derivative of the total mechanical energy simplifies to

\dot T_{r+e}(t) =<br /> \frac 1 2 \dot m_e(t) v_e(t)^2<br />

https://www.physicsforums.com/showthread.php?t=199087

 

[trivia1]

The following, regardless of the KE of the spent fuel being ignored in the calculation, is correct.

A rocket in space, of mass 1kg, accelerates at 1m/s squared. Between t=0 and t=1 it's change in KE is 0.5j, between t=999 and t=1000 it's change in KE is 999.5j. KE is relative to an observer stationary at t=0.

I’m assuming the engine applying the force is doing so by firing out minute particles at extremely high velocities, so as any loss in rocket mass –that is unspent fuel- is negligible. This is an ideal scenario in a Newtonian universe, so the particles can be as small as we wish them to be and the velocities as high as we decide. It just has to be consistent with Newton’s physics.
Gaining an understanding of the physics of rocket propulsion –interesting as it is- is not why I posted. It’s the unintuitive nature of the relationship between KE and V squared that I was trying to grasp. Bob for Short has explained that Power is relative to Velocity. That would mean that for an observer at rest in relation to the Rocket at t=0, at t=1000 that rocket’s engine is massively more powerful. That seems more than a little odd, but it does give me a new perspective. It means that an engine exerting a force on a mass, whether in deep space or on a road on Earth, is more powerful the higher it’s velocity. To a layperson in physics, that’s plain strange.
Is it correct to state that as an engine’s velocity increases, it’s power output in reference to an initial observer increases exponentially?

 

[A.T.]

It means that an engine exerting a force on a mass, whether in deep space or on a road on Earth, is more powerful the higher it’s velocity.

power = force * velocity

 

[qraal]

The following, regardless of the KE of the spent fuel being ignored in the calculation, is correct.

A rocket in space, of mass 1kg, accelerates at 1m/s squared. Between t=0 and t=1 it's change in KE is 0.5j, between t=999 and t=1000 it's change in KE is 999.5j. KE is relative to an observer stationary at t=0.

I’m assuming the engine applying the force is doing so by firing out minute particles at extremely high velocities, so as any loss in rocket mass –that is unspent fuel- is negligible. This is an ideal scenario in a Newtonian universe, so the particles can be as small as we wish them to be and the velocities as high as we decide. It just has to be consistent with Newton’s physics.
Gaining an understanding of the physics of rocket propulsion –interesting as it is- is not why I posted. It’s the unintuitive nature of the relationship between KE and V squared that I was trying to grasp. Bob for Short has explained that Power is relative to Velocity. That would mean that for an observer at rest in relation to the Rocket at t=0, at t=1000 that rocket’s engine is massively more powerful. That seems more than a little odd, but it does give me a new perspective. It means that an engine exerting a force on a mass, whether in deep space or on a road on Earth, is more powerful the higher it’s velocity. To a layperson in physics, that’s plain strange.
Is it correct to state that as an engine’s velocity increases, it’s power output in reference to an initial observer increases exponentially?

Power increases, from one point of view, but from the rocket's and the exhaust's perspective nothing changes - so long as the mass-flow is negligible. Power is the time derivative of energy. As a rocket gets faster so too its kinetic energy gets higher, but at the square of the velocity. So you'd expect the power, as time derivative (the instantaneous rate of change) of energy to increase quickly too, in step with the energy. There's nothing really strange about that is there?

 

[D H]

A rocket in space, of mass 1kg, accelerates at 1m/s squared. Between t=0 and t=1 it's change in KE is 0.5j, between t=999 and t=1000 it's change in KE is 999.5j. KE is relative to an observer stationary at t=0.

Gaining an understanding of the physics of rocket propulsion –interesting as it is- is not why I posted. It’s the unintuitive nature of the relationship between KE and V squared that I was trying to grasp. Bob for Short has explained that Power is relative to Velocity. That would mean that for an observer at rest in relation to the Rocket at t=0, at t=1000 that rocket’s engine is massively more powerful. That seems more than a little odd, but it does give me a new perspective. It means that an engine exerting a force on a mass, whether in deep space or on a road on Earth, is more powerful the higher it’s velocity. To a layperson in physics, that’s plain strange.
Is it correct to state that as an engine’s velocity increases, it’s power output in reference to an initial observer increases exponentially?

One explanation for this apparent paradox is that energy is a frame dependent concept. During that first second of operation, an observer initially at rest with respect to the rocket will see your rocket's energy change by 0.5 joules. Compare this to the change in energy as observed from the perspective of someone moving toward the accelerating rocket with a constant velocity of 999 m/s: A change of 999.5 joules.

An even better explanation, IMO, is that the paradox arises from looking at an open system. This view can make it look like rockets violate conservation of energy. One answer to this: There is of course no expectation that energy is conserved in an open system.

When you look at the rocket+exhaust cloud as a whole, the change in energy *is* frame-independent. A rocket in a pure vacuum and far removed from any gravitational bodies is not subject to any external forces. The rocket combined with the exhaust cloud the rocket leaves behind form an isolated system. The only change in the kinetic energy of the rocket+exhaust cloud system arises from converting the potential energy of the fuel to kinetic energy, and this change is frame-independent.

 

[Dadface]

trivia, forget rockets for a moment and look at another simpler case which might appeal to your intuition,an object in free fall in a vacuum.In this case the force on the object remains constant, its kinetic energy increase in the way you expect it to but then so does its power increase and all the sums add up.In this and other examples the force remains constant but the power(Fv) changes.
Rockets seem to be an exception to this and we have been told that with these that if the force remains constant the power remains constant also"the KE of the exhaust gases plus rocket increase at a constant rate".In my opinion this makes rockets really interesting and like yourself I am struggling a bit to get this concept in my head in an intuitive(and in my case non mathematical) way.I think I am beginning to gain an understanding but I need to look again at the excellent tutorials posted by DH.It's the hard stuff and the stuff which promotes discussion which makes physics such an interesting subject.

 

[Dale]

regardless of the KE of the spent fuel being ignored in the calculation

trivia1, it is really annoying, and frankly pretty rude, for you to come here and ask a question and then deliberately ignore the answer you received multiple times from multiple people. If you were not interested in hearing the responses and learning the resolution to your problem then why did you bother to post the question in the first place? All you have accomplished is to waste our time and make us feel foolish for bothering to try to help someone who didn't really want our help.

 

[trivia1]

trivia1, it is really annoying, and frankly pretty rude, for you to come here and ask a question and then deliberately ignore the answer you received multiple times from multiple people. If you were not interested in hearing the responses and learning the resolution to your problem then why did you bother to post the question in the first place? All you have accomplished is to waste our time and make us feel foolish for bothering to try to help someone who didn't really want our help.

By ignoring the KE of the spent fuel I hoped to simplify the model to gain some sort of insight. Bob_for_short explained the increase in KE by referring to the force acting over distance, which also ignores the spent fuel.
I'm not deliberately ignoring answers, I've read the posts, I simply don't get it intuitively. Mathematically it makes sense, but the concepts the maths support are difficult for me to grasp.
Why anyone would feel foolish for trying to answer a question is also a puzzle to me. If you've answered it, and I haven't got it, why feel foolish? People continue to post answers, that go along the same lines as the first, because they make the perfectly reasonable and correct assumption that it’s a genuine puzzle for me, and they know I’m not deliberately ignoring anything. If you don’t make that assumption, don’t post.

 

[Dale]

If you ignore the exhaust then your system is not closed and energy is not conserved.

 

[D H]

By ignoring the KE of the spent fuel I hoped to simplify the model to gain some sort of insight. Bob_for_short explained the increase in KE by referring to the force acting over distance, which also ignores the spent fuel.

Bob_for_short did a disservice. Rockets work by transferring momentum to space, and they do this by ejecting mass. Ignore that and you will come up with an apparent paradox. The paradox vanishes as soon as you realize

• The rocket by itself is an open system. Open systems do not in general conserve mass, linear momentum, angular momentum, or energy. Combining this with the fact that energy is frame-dependent leads to a big fat "So what!"
• OR you could look at the rocket+exhaust as a closed system. The paradox disappears: mass, momenta, and energy are all conserved.

However, a new apparent paradox appears in the form of Tsiolkovsky rocket equation. You want to accelerate a rocket from rest to three times the exhaust velocity? No problem. You just need the initial mass (vehicle+fuel) to be 95% fuel. Six times the exhaust velocity? Big problem. Now the initial mass needs to be 99.75% fuel.

 

[rcgldr]

The other issue you're ignoring is the point of application of that force. A car applies a force to the pavement it moves on, an airplane applies a force to the air it moves through, in these cases, the power is related to the force times speed at the point of application of that force. However a rocket doesn't interact with the space it moves through. Instead the force is generated internally, by expelling a bit of itself backwards at high speed. In this case the point of application of force is at the rocket nozzle, any remaining onboard fuel is accelerated along with the rocket, and the power generated is a function of how much and how fast the onboard fuel is accelerated, not related to the rockets speed relative to some other object or frame of reference in space.

 

[qraal]

The other issue you're ignoring is the point of application of that force. A car applies a force to the pavement it moves on, an airplane applies a force to the air it moves through, in these cases, the power is related to the force times speed at the point of application of that force. However a rocket doesn't interact with the space it moves through. Instead the force is generated internally, by expelling a bit of itself backwards at high speed. In this case the point of application of force is at the rocket nozzle, any remaining onboard fuel is accelerated along with the rocket, and the power generated is a function of how much and how fast the onboard fuel is accelerated, not related to the rockets speed relative to some other object or frame of reference in space.

Doesn't matter. The rules still rule regardless of the reference frame one picks. What doesn't work is comparing a measurement in one reference frame against a measurement in another and then wondering why they apparently disagree.

 

[FoxCommander]

A rocket in space, of mass 1kg, accelerates at 2m/s squared. Between t=0 and t=1 it's change in KE is 0.5j, between t=999 and t=1000 it's change in KE is 999.5j. Yet the rocket motor power output hadn't changed. What explains the massive difference in KE transferred to the rocket?

well consider the speed that the object is going at t=999, it is going 1998 m/s. which is pretty fast. the amount of work done by the rocket is the force of the rocket * the distance traveled. If it is going that fast for one second, not even counting acceleration, its distance will be 1998 meters (with acceleration it turns out to be 1999), that's almost 2Km in just 1 second! Now multiply that to the force, which is 2N and you get 3998Joules. which is also a very big amount of energy, for the size.

You can see why even a small force over a great distance can become a lot of energy. this is why the difference is so big.