What is the role of external forces in rocket propulsion?

[Oblio]

I have this equation:

dP = P(t+dt) - P(t) = mdv +dmv(ex)

(ex=external)

I have to show that that will equal

ma = -ma + F(ex)

(Although in the book, acceleration is shown as the derivative of v, (dotted v)).

In the example I was reading, there are no external forces, so my first equation simply boiled down to mdv=-dmv.
I'm not sure how to substitute in ALL 3 of those momentums above...

Thanks!

 

[Oblio]

My bad!

Sorry. V(ex) is exhaust speed at which the rocket is ejecting spent fuel, relative to the rocket.

 

[Oblio]

In further reading, I'm not sure about the earlier stage of the first equation I posted...

A rocket is traveling in the positive x with mass m,and velocity v, and ejecting spent fuel at exhaust speed of v(ex) relative to the rocket. Since the rocket is ejecting mass, the rockets mass m is steadily decreasing. At time t, the momentumis P(t) = mv. A short time later at t+dt the rockets mass is m+dm where dm is negative, and its momentum is (m+dm)(v+dv). The fuel ejected in the time dt has mass (-dm) and velocity v-v(ex) relative to the ground. thus the total momentum (rocket plus the fuel just ejected) at t+dt is

P(t+dt)= (m+dm)(v+dv) - dm(v-v(ex)

If dm is already negative, why is it subtracted again?

 

[Astronuc]

The mass of propellant being expelled is the source of the force/momentum applied to the rocket, but it also means less mass to be accelerated or subjected to that force.

The remaining propelleant is a mass, in addition to the payload mass, that must be accelerated.

 

[Oblio]

Ah ok.
So the first (m+dm) is the 'cargo' propellant, and the '-dm(v-v(ex)) is my external force?

 

[Oblio]

In one way I think it has to be F (ext) but dm is just v... there's no acceleration in those terms...

 

[Oblio]

when the book has -mdv =dmv (ex) the simplify by dividing both by dt...

but I have -mg = mdv + dmv (ex).

I can't find a way to use separation of variables here...

 

[Oblio]

Although this is done and handed in, I'm still not sure about:

P(t+dt)= (m+dm)(v+dv) - dm(v-v(ex)

is dm(v-vex) my external force?

 

[learningphysics]

Can you write the question word for word? I don't understand what you need to prove.

 

[Oblio]

Can you write the question word for word? I don't understand what you need to prove.

With the help of the book I completed the question, but didn't understand steps along the way that the book made.

Anyways,

Consider a rocket traveling in a straight line subject to an external force F(ext) acting along the same line. Show that its equation of motion is m$$\dot{v}$$ = -$$\dot{m}$$v(ex) + F(ext).
It says to review the derivation of this:

m$$\dot{v}$$=-$$\dot{m}$$v (ex) (eqn 3.6)

from

P(t+dt) = (m+dm)(v+dv)-dm(v-v(ex)) = mv+dmv + dmv(ex) (he neglects 'the doubly small product dmdv)

which turns into:

dP=P(t+dt) - P(t) = mdv+dmv(ex) (eqn 3.4) and this turns into 3.6




So that in my case, I am keeping the external force, where they got rid of it to get 3.6.

So I set it up to be:

-mg=mdv +dmv(ex)



In the very beginning,
am I correct in saying that: -dm(v-v(ex)) is my external force?

 

[D H]

No. The external force is something generated by some external agent (hence the name), such as the atmosphere. Forces are additive, so you can add the effective force from thrusting and the external force to get the total force.

 

[learningphysics]

The external force just equals $$\frac{dP}{dt}$$. in your book example external force is 0, hence dP/dt is 0. But in this case external force is nonzero.

 

[Oblio]

P(t+dt) = (m+dm)(v+dv)-dm(v-v(ex))

what IS -dm(v-vex) then?

P=mv, and that exists before this part of the equation..

 

[learningphysics]

P(t+dt) = (m+dm)(v+dv)-dm(v-v(ex))

what IS -dm(v-vex) then?

P=mv, and that exists before this part of the equation..

-dm(v-vex) is the momentum of the exhaust.

P(t+dt) - P(t) = F(ext)*dt

Don't get vex confused with "external"... vex is the velocity of the exhaust relative to the rocket.