1. Some of the following is more or less Halliday & Resnick, 4th Ed., section 9.8 (not the whole section and I added a lot):

Assume motion in a stright line. Therefore, for position, velocity, acceleratiom, force, momentum, etc., d/dt|vector|=|d/dt(vector)|. This is not true in general for motion along a curved path. Assume we have an idealized rocket of mass m and speed |v| at time t traveling in the forward direction relative to an inertial reference frame. The rocket then instantaneously begins to emit a constant stream of exhaust in the backwards direction.

let:
F_net_ext = net external force = net force on (rocket + exhaust) = sum of gravity, atmospheric drag, etc.
F_net_r = net force on the rocket
P = momentum of (rocket + exhaust)
p = momentum of rocket
u = velocity of exhaust (as it leaves the rocket) pointing backwards.
v_rel = velocity of exhaust (as it leaves the rocket) relative to rocket = u - v = vector pointing back and is constant for an idealized rocket.
Note F_net_ext, F_net_r, P, p, v, u, v_rel, dv, a, etc. are all vetors. Their lengths are |vector|.

In time interval dt the rocket emits a mass -dm. Note that dm < 0 and |dm/dt| = constant for an idealized rocket = mass flow rate of exhaust backwards from the rocket. So at time t+dt we now have a bit of exhaust of mass -dm moving backwards with velocity u and the rocket, now of mass m+dm, moving forward at velocity v+dv. Note dv clearly points forward: the rocket's acceleration, a, points forward.

(1) F_net__ext = dP/dt = [1/dt)][(m+dm)(v+dv)+(-dm)u - mv] = mdv/dt + vdm/dt - udm/dt (note (dv)dm/dt ->0 in the limit of the derivative)

This equation is standard for rockets. There are generally 2 ways it is rewritten:

(2) F_net_ext = mdv/dt + vdm/dt - udm/dt = dp/dt - udm/dt
(3) F_net_ext = mdv/dt + vdm/dt - udm/dt = mdv/dt - (u-v)dm/dt = ma - (v_rel)dm/dt

F_net_ext is not the force that propels the rocket. The force that propels the rocket is an internal force within the (rocket+exhaust) system. If F_net_ext = 0 then P remains constant but the rocket experiences a thrust which changes its momentum. The change in p in time dt is equal and opposite to the momentum, udm, carried away by the exhaust. for a rocket (u-v)dm/dt = thrust = (v_rel)dm/dt = forward pointing vector since u-v points back and dm/dt < 0. The thrust is the rate at which momentum enters the rocket. For an idealized rocket with constant v_rel and constant dm/dt, the thrust is a constant.

This is pretty much the extent of Halliday & Resnick on this subject, but every time I read it I'm struck by the seeming inconsistency that thrust is clearly constant yet the rate of change of the rocket's momentum, dp/dt, is not.

I hope you wont mind if I walk you thru my reasoning. By (3), F_net_ext=0 implies ma=thrust. thrust is constant and points forward implies ma is therefore a forward pointing constant vector. ma is constant and m is decreasing with time implies |a| is increasing with time. Therefore as long as thrust exits the rocket backwards, then the rocket will not only accelerate in the forward direction, but the rate at which it accelerates increases with time. dm/dt < 0 and v always points forward implies the term vdm/dt is a backwards pointing vector which is not constant but grows in length as v grows. At time t_0 when the exhaust initially begins to flow back from the rocket, u is a backwards pointing vector of length |v_rel| since v=0. u then shrinks in length as the rocket moves faster, but v_rel remains constant. At some point |v| = |v_rel| and u=0, and after this |v| > |v_rel| and u points forward, tho still shorter than v by |v_rel|. By (2), F_net_ext=0 implies dp/dt = udmdt. dm/dt<0 implies the term udm/dt points forward until |v|>|v_rel|, then it points back. Therefore dp/dt points forward and then, after |v|>|v_rel|, dp/dt points back.

This is one of the major counterintuitive things about rockets that confuses me (and presumably most people, unless my calculations are incorrect). When |v|>|v_rel|, dp/dt of the rocket points back, meaning the net force on the rocket (not F_net_ext) points backwards, opposite the forward motion of the rocket. The momentum of the rocket always points forward in this example, but it's getting shorter at the rate of dp/dt, and tho the net force on the rocket points back after |v| exceeds |v_rel|, nevertheless the rocket not only continues to accelerate forward, but the rate of forward acceleration increases with time!

Many books and websites claim that, in the absence of external forces such as gravity, friction, etc., the net force on a rocket is equal to the thrust. I've even seen this on some NASA websites. But the net force on a rocket does not seem to be equal to the thrust. Assuming F_net_ext = 0 then F_net_r = dp/dt = udm/dt by (2) = udm/dt + vdm/dt - vdm/dt = (u-v)dm/dt + vdm/dt = thrust + vdm/dt. So the net force on a rocket, absent external forces, does not seem to be equal to the thrust, but rather equal to the thrust + vdm/dt.

Hopefully, at this point, you're saying to yourself "what's the problem?" If so, then there probably is no problem. But I'm bothered by the equations:

(4) F_net_ext=0 implies F_net_r = thrust + vdm/dt = u dm/dt

Both equations seem counterintuitive even tho I've reasoned thru the details. It bothers me that many books and websites set F_net_r = thrust + external forces and ignore the vdm/dt term. It makes me wonder whether eqn (4) is correct. I haven't found either of the equations (4) in Halliday & Resnick or Marion & Thornotn's Classical Dynamics or Goldstein's Classical Mechanics, but I haven't found anything in those texts that disputes eqns (4) either.

I would like to know if eqns (4) are correct? Is there something simple here that I'm missing? Thanks in advance.

2.

3. Originally Posted by inkliing
Some of the following is more or less Halliday & Resnick, 4th Ed., section 9.8 (not the whole section and I added a lot):

Assume motion in a stright line. Therefore, for position, velocity, acceleratiom, force, momentum, etc., d/dt|vector|=|d/dt(vector)|. This is not true in general for motion along a curved path. Assume we have an idealized rocket of mass m and speed |v| at time t traveling in the forward direction relative to an inertial reference frame. The rocket then instantaneously begins to emit a constant stream of exhaust in the backwards direction.

let:
F_net_ext = net external force = net force on (rocket + exhaust) = sum of gravity, atmospheric drag, etc.
F_net_r = net force on the rocket
P = momentum of (rocket + exhaust)
p = momentum of rocket
u = velocity of exhaust (as it leaves the rocket) pointing backwards.
v_rel = velocity of exhaust (as it leaves the rocket) relative to rocket = u - v = vector pointing back and is constant for an idealized rocket.
Note F_net_ext, F_net_r, P, p, v, u, v_rel, dv, a, etc. are all vetors. Their lengths are |vector|.

In time interval dt the rocket emits a mass -dm. Note that dm < 0 and |dm/dt| = constant for an idealized rocket = mass flow rate of exhaust backwards from the rocket. So at time t+dt we now have a bit of exhaust of mass -dm moving backwards with velocity u and the rocket, now of mass m+dm, moving forward at velocity v+dv. Note dv clearly points forward: the rocket's acceleration, a, points forward.

(1) F_net__ext = dP/dt = [1/dt)][(m+dm)(v+dv)+(-dm)u - mv] = mdv/dt + vdm/dt - udm/dt (note (dv)dm/dt ->0 in the limit of the derivative)

This equation is standard for rockets. There are generally 2 ways it is rewritten:

(2) F_net_ext = mdv/dt + vdm/dt - udm/dt = dp/dt - udm/dt
(3) F_net_ext = mdv/dt + vdm/dt - udm/dt = mdv/dt - (u-v)dm/dt = ma - (v_rel)dm/dt

F_net_ext is not the force that propels the rocket. The force that propels the rocket is an internal force within the (rocket+exhaust) system. If F_net_ext = 0 then P remains constant but the rocket experiences a thrust which changes its momentum. The change in p in time dt is equal and opposite to the momentum, udm, carried away by the exhaust. for a rocket (u-v)dm/dt = thrust = (v_rel)dm/dt = forward pointing vector since u-v points back and dm/dt < 0. The thrust is the rate at which momentum enters the rocket. For an idealized rocket with constant v_rel and constant dm/dt, the thrust is a constant.

This is pretty much the extent of Halliday & Resnick on this subject, but every time I read it I'm struck by the seeming inconsistency that thrust is clearly constant yet the rate of change of the rocket's momentum, dp/dt, is not.

I hope you wont mind if I walk you thru my reasoning. By (3), F_net_ext=0 implies ma=thrust. thrust is constant and points forward implies ma is therefore a forward pointing constant vector. ma is constant and m is decreasing with time implies |a| is increasing with time. Therefore as long as thrust exits the rocket backwards, then the rocket will not only accelerate in the forward direction, but the rate at which it accelerates increases with time. dm/dt < 0 and v always points forward implies the term vdm/dt is a backwards pointing vector which is not constant but grows in length as v grows. At time t_0 when the exhaust initially begins to flow back from the rocket, u is a backwards pointing vector of length |v_rel| since v=0. u then shrinks in length as the rocket moves faster, but v_rel remains constant. At some point |v| = |v_rel| and u=0, and after this |v| > |v_rel| and u points forward, tho still shorter than v by |v_rel|. By (2), F_net_ext=0 implies dp/dt = udmdt. dm/dt<0 implies the term udm/dt points forward until |v|>|v_rel|, then it points back. Therefore dp/dt points forward and then, after |v|>|v_rel|, dp/dt points back.

This is one of the major counterintuitive things about rockets that confuses me (and presumably most people, unless my calculations are incorrect). When |v|>|v_rel|, dp/dt of the rocket points back, meaning the net force on the rocket (not F_net_ext) points backwards, opposite the forward motion of the rocket. The momentum of the rocket always points forward in this example, but it's getting shorter at the rate of dp/dt, and tho the net force on the rocket points back after |v| exceeds |v_rel|, nevertheless the rocket not only continues to accelerate forward, but the rate of forward acceleration increases with time!

Many books and websites claim that, in the absence of external forces such as gravity, friction, etc., the net force on a rocket is equal to the thrust. I've even seen this on some NASA websites. But the net force on a rocket does not seem to be equal to the thrust. Assuming F_net_ext = 0 then F_net_r = dp/dt = udm/dt by (2) = udm/dt + vdm/dt - vdm/dt = (u-v)dm/dt + vdm/dt = thrust + vdm/dt. So the net force on a rocket, absent external forces, does not seem to be equal to the thrust, but rather equal to the thrust + vdm/dt.

Hopefully, at this point, you're saying to yourself "what's the problem?" If so, then there probably is no problem. But I'm bothered by the equations:

(4) F_net_ext=0 implies F_net_r = thrust + vdm/dt = u dm/dt

Both equations seem counterintuitive even tho I've reasoned thru the details. It bothers me that many books and websites set F_net_r = thrust + external forces and ignore the vdm/dt term. It makes me wonder whether eqn (4) is correct. I haven't found either of the equations (4) in Halliday & Resnick or Marion & Thornotn's Classical Dynamics or Goldstein's Classical Mechanics, but I haven't found anything in those texts that disputes eqns (4) either.

I would like to know if eqns (4) are correct? Is there something simple here that I'm missing? Thanks in advance.

Rockets are accelerating and of variable mass. Therefore F=ma does not apply. The momentum of a rocket, in an inertial reference frame does not always increase as the rocket accelerates, because the mass is continually decreasing. At the point at which the speed of the rocket, relative to the inertial frame, exceeds the speed of the exhaust gasses, relative to the rocket, both the gas and the rocket have velocity vectors with the same direction, relative to the inertial frame, and the momentum of the rocket begins to decrease even as the speed increases.

Force relative to the inertial frame is but needs to be treated carefully. The speed of the rocket relative to an inertial frame, in the absence of gravity and drag is given by the "rocket equation" which can be derived from considerations of conservation of momentum:

Isp s the speed of the exhaust gasses relative to the rocket at the nozzle exit plane.

4. Please don't quote in entirety such a lengthy post unless it's necessary. It wastes space.

Originally Posted by DrRocket
Rockets are accelerating and of variable mass. Therefore F=ma does not apply. The momentum of a rocket, in an inertial reference frame does not always increase as the rocket accelerates, because the mass is continually decreasing. At the point at which the speed of the rocket, relative to the inertial frame, exceeds the speed of the exhaust gasses, relative to the rocket, both the gas and the rocket have velocity vectors with the same direction, relative to the inertial frame, and the momentum of the rocket begins to decrease even as the speed increases.
This is a restatement of part of my post. I don't know why you felt it was necessary to restate what I already said.

Force relative to the inertial frame is but needs to be treated carefully. The speed of the rocket relative to an inertial frame, in the absence of gravity and drag is given by the "rocket equation" which can be derived from considerations of conservation of momentum:

Isp s the speed of the exhaust gasses relative to the rocket at the nozzle exit plane.
The rocket equation proceeds straightforwardly from ma=thrust, eqn (2) with f_net_ext=0. This has nothing to do with the question I asked.

5. Originally Posted by inkliing
The rocket equation proceeds straightforwardly from ma=thrust, eqn (2) with f_net_ext=0. This has nothing to do with the question I asked.
No, it does not, since thrust. Thrust , thrust being by definition the net force on the rocket as measured in an inertial frame. The component is only part of the story.

Further it has a great deal to do with your question, except that you don't understand your own question.

The problem comes from attempting to take data from static test firings and applying it naively to a rocket in flight. You cannot do that because the rocket is both accelerating, hence an attached reference frame is not inertial, and since the massis constantly changing.

If you work out the mechanics carefully in an inertial frame, it takes a bit of work, then all is well.

BTW if you want to lecture someone, try someone else. I am fairly confident that I have been in volved in the design and manufacture of quite a few more rockets than have you.

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