- #1

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- Homework Statement
- A rocket is in a space. The rocket's initial mass and velocity is m and v. After that, mass dm is ejected backwards with constant speed u relative to the rocket. Find the equation for the velocity of the rocket

- Relevant Equations
- momentum conservation

I have a question. If we assume that ##dm## is positive, is the answer supposed to be different from the one when we regard the ##dm## as negative?

By using momentum conservation, we will get

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

simplify the equation

$$m \,dv=dm \,u$$

Integrate the equation and we will get

$$ln\frac {m'} {m} = \frac {v'-v} {u}$$

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

Simplify

$$m \,dv=-dm \,u$$

The answer will be

$$ln\frac {m'} {m} = - \frac {v'-v} {u}$$

What exactly happened here?

**1. If I assume that ##dm## is positive:**By using momentum conservation, we will get

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

simplify the equation

$$m \,dv=dm \,u$$

Integrate the equation and we will get

$$ln\frac {m'} {m} = \frac {v'-v} {u}$$

**2. if I assume that mass ##dm## is negative:**$$mv=(m+dm)(v+dv)-dm (v-u)$$

Simplify

$$m \,dv=-dm \,u$$

The answer will be

$$ln\frac {m'} {m} = - \frac {v'-v} {u}$$

What exactly happened here?