Rocket motion - conservation of momentum

[bensondros]

Homework Statement

Find the speed v of the rocket when the mass of the rocket = m. The rocket starts from rest at with mass M. Fuel is ejected at speed u relative to the rocket.

Homework Equations

m₁v₁ = m₂v₂

The Attempt at a Solution

In the textbook, it starts off with a moving rocket with mass m and speed v. The fuel is given the mass (-dm) which is positive. So after a short time dt, the mass of the rocket changes to m+dm and speed v+dv. The mass of the ejected fuel is (-dm) and since it ejects with speed u relative to the rocket traveling at speed v, the ejected fuel travels at speed v-u, which can be positive or negative depending on which of v or u is larger.

Writing down the equation of conservation of momentum:

mv = (m+dm)(v+dv) + (-dm)(v-u)

which then leads to

m dv = -u dm

After a few steps, we get:

v₂-v₁ = u ln(m₁/m₂)


That all looks fine and understandable. However, it mentions in the book that I am free to define dm to be positive, and then subtract it from the rocket's mass, and have dm get shot out the back. So I have decided to try it.

Writing down the equation of conservation of momentum:

mv = (m-dm)(v+dv) + dm(v-u)

which then leads to

m dv = u dm (note that at this point, the equation is already different from before)

Moving the variables around and integrating v from v₁ to v₂, m from m₁ to m₂ as before, I obtained:

v₂-v₁ = u ln(m₂/m₁)

which is clearly wrong because m₂<m₁, so

ln(m₂/m₁) < 0

but

v₂ > v₁


So what went wrong there?

 

[Delphi51]

Well, thanks for an evening's entertainment! I could not wrap my mind around it. Clearly dm has to be negative in order for the value of m to diminish. But it is natural to think of dm as positive when we write (m-dm). Conflict! I found clarification here
http://ed-thelen.org/rocket-eq.html
Some explanations use one m for the rocket and another m for the propellant.
Anyway, it seems one must use different letters for the m in dm and the M for mass of the rocket. Then dM = - dm and it all works out.

How clever of the author of your book to make us really think!