(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

m_{1}v_{1}= m_{2}v_{2}

3. 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 travelling 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:

v_{2}-v_{1}= u ln(m_{1}/m_{2})

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 v_{1}to v_{2}, m from m_{1}to m_{2}as before, I obtained:

v_{2}-v_{1}= u ln(m_{2}/m_{1})

which is clearly wrong because m_{2}<m_{1}, so

ln(m_{2}/m_{1}) < 0

but

v_{2}> v_{1}

So what went wrong there?

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# Rocket motion - conservation of momentum

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