Solve Counterintuitive Space Vehicle Problem w/Two Thrusters

[DaveC426913]

Once the fuel starts flowing to the left, the ship has to move to the right in order to keep the momentum of the system zero. The velocity of the exhaust has no horizontal component, i.e., is completely vertical (up and down) with repect to the ship. Because the ship is moving to the right, the exhaust moves (partly) to the right. After the thrusters are turned off, fuel no longer flows, but the exhaust is still moving a bit to the right. In order for the momentum of the system to be zero, the ship has to be drifting to the left.

I hope I'm not beating this to death. (And I don't doubt that D_H and you are correct.) But I really want to understand (I hate not understanding).

I understand the rationale in the above explanation (the overall momentum must be zero). But I want to understand it "from the inside" i.e. from the frame of ref of the occupants. It would have to be an internally consistent cause and effect. Somehow, the exhaust plume must be observed as slightly rightward-of-normal - in order for the ship to ultimately have a slightly leftward movement.

 

[D H]

I understand the rationale in the above explanation (the overall momentum must be zero). But I want to understand it "from the inside" i.e. from the frame of ref of the occupants. It would have to be an internally consistent cause and effect. Somehow, the exhaust plume must be observed as slightly rightward-of-normal - in order for the ship to ultimately have a slightly leftward movement.

At the end of the day, I would expect to see two backwards letter Cs drifting away from each other tranvsersely. The tip and tail of each C will be stationary longitudinally. The Cs will stretch out as time passes.

The equations of motion for this system, assuming continuous thruster firings, are

m_v(t)\ddot r_v(t) + \ddot m_v(t) l = 0

where l is the vector from the midpoint of the two thrusters and the fuel tank.

The derivation is left to the reader.

The vehicle accelerates only when the flow rate changes. The exhaust plumes will replicate the graph of the flow rate.

 

[DaveC426913]

OOHHHH!

Upon turning on the jets, the craft accelerates to the right.
Upon turning off the jets, the craft accelerates to the left BUT the craft is slightly less massive than before, causing it to have an slightly GREATER acceleration leftwards!

Jeez. Why didn't you just say so!

Izzat right? I hope that's right.

 

[D H]

That is exactly correct.

 

[DaveC426913]

I wonder if I can make a career out of lay-interpreting scientific explanations.

 

[Jheriko]

I wonder if I can make a career out of lay-interpreting scientific explanations.

From what I can tell you don't actually have to be any good at it to make money from it.

 

[reilly]

Great problem. I wish I'd known it when I was terrorizing graduate students in their physics PhD orals. Another great one, supposedly originating with Enrico Fermi, is: how far can a bird fly?

Regards,
Reilly Atkinson

 

[DaveC426913]

African or European?

 

[daniel_i_l]

OOHHHH!

Upon turning on the jets, the craft accelerates to the right.
Upon turning off the jets, the craft accelerates to the left BUT the craft is slightly less massive than before, causing it to have an slightly GREATER acceleration leftwards!

Jeez. Why didn't you just say so!

Izzat right? I hope that's right.

One thing i don't inderstand about this - in the first step (turning on the jets) the craft and the exhaust are accelerating to the right. but since there're no external forces how can the "system" have a positive momentum?

 

[D H]

The "system" does not have a positive momentum. It has zero momentum. The "system" comprises the solid vehicle, the fuel in the tank, the exhaust, and the fuel flowing inside the vehicle from the tank to the thrusters. It is the leftward momentum of that fuel flow that counterbalances the rightward momentum of everything else.

 

[daniel_i_l]

Ok, i hope I'm not being stupid here but I've thought about it awhile and am convinced that after the thrusters are turned off the craft will have moved to the right but will have no speed, and the exhaust will also have 0 speed, here's why:
instead of fuel in the pipe think about someone throwing a ball (really 2 balls, one for each thruster from the place that the fual tank is to a "reflector" (a slanted wall) right under the thruster like this: (the ">" in the middle are the 2 reflectors)
+=====v=========+
|. . . . . .[/color]. . . . . . . . [/color]|
|. . . . . [/color]>-----O . . . [/color]|
|. . . . . .[/color]. . . . . . . . [/color]|
+=====^=========+
now let's see what happens when the person throws one ball at a reflector- first the craft is moving to the right and the ball to the left, then when the ball hits the reflector the craft and ball stop moving and the ball flies up - notice that I'm not using conservation of kinetic energy, i think that this is reasonable for liquid hitting a wall. so throwing one ball just moves the craft over a little to the right.
now if the person is throwing a lot of balls one after the other at a constant rate then the craft will stop moving because as soon as one ball is thrown another one hits the reflector. but if the rate of the throws are accelerated then the craft will start moving to the right because more balls are being thrown then are hitting the reflector at any given time interval. but if the person stops throwing the balls then all the balls that are currently between the person and reflector eventually hit the reflector and the craft stops.
so in the end the craft will haved moved to the right and the balls will all be flying straight up and to the left of the craft - each by a different amount.
what is wrong with this reasoning?
Thanks.

 

[D H]

Your reasoning is correct only if at most one ball is in flight at any time between the thrower and the reflector. Now suppose the thrower releases balls in rapid-fire mode such that multiple balls are in flight between the thrower and reflector. I will assume the thrower fires balls at a constant rate and at a constant relative velocity and continues to do so for some time.

The vehicle starts at rest with no balls in flight. The thrower starts firing balls. The vehicle's velocity changes with each thrown ball up to the time the first ball hits the reflector. When the first ball hits the reflector, it no longer stops the vehicle because there are N-1 balls in flight behind it. Instead, the collision merely reduces the vehicle's velocity a bit. The ball exits the vehicle with the vehicle's non-zero longitudinal velocity.

 

[daniel_i_l]

Hmm... it seems like the real problem in my logic was the fact that the ball bounced straight up instead of up and to the right, if it actually bounces to the right then the craft would be deflected a little to the left and so in the end it's position would be moved to the right but it would have a velocity to the left. is that correct?

 

[D H]

The balls bounce straight up/down---relative to the vehicle, that is. The have to, for two reasons:
(1) They will miss the exit holes in your diagram¹.
(2) Anything else won't re-create the original problem.

¹This ignores the problem of longitudinal acceleration during the flight from the reflector to the exit hole.

 

[DaveC426913]

but if the person stops throwing the balls then all the balls that are currently between the person and reflector eventually hit the reflector and the craft stops.

There's one piece you're forgetting here.

With a tank that's now half empty of balls, the craft is less massive than when you started. A ball throw with the same force will accelerate the craft more than at first. (a=F/m: as m decreases, a increases)