I The rocket equation, one more time

[Rick16]

TL;DR

Trying to understand the setup of the rocket equation

I already posted a similar thread a while ago, but this time I want to focus exclusively on one single point that is still not clear to me.

I just came across this problem again in Modern Classical Mechanics by Helliwell and Sahakian. Their setup is exactly identical to the one that Taylor uses in Classical Mechanics: a rocket has mass m and velocity v at time t. At time $t+\Delta t$ it has (according to the textbooks) velocity $v + \Delta v$ and mass $m+\Delta m$. Why not $m - \Delta m$? This is all the more strange since just 2 pages further on Helliwell and Sahakian give the example of an open railroad boxcar moving in the rain, which accumulates a mass $\Delta m_r$ due to the rain falling in, and which loses mass $\Delta m_l$ due to water leaking out. Then they write "... at time $t+\Delta t$ ... boxcar of mass $M+\Delta m_r - \Delta m_l$, indicating that the boxcar has gained mass $\Delta m_r$ due to the falling rain, while losing mass $\Delta m_l$ due to the leak."

So in the case of the boxcar they write $m - \Delta m$ to indicate a decreasing mass, whereas in the case of the rocket they write $m + \Delta m$ to indicate the exact same situation. How can one make sense of this?

 

[PeroK]

$\Delta m$ is generally defined as the change in $m$. If the mass reduces, then $\Delta m$ is negative.

It's a common point of confusion to think that negative quantities must be subtracted. In general, we write $m + \Delta m$ regardless of whether $\Delta m$ is negative or positive.

 

[Rick16]

$\Delta m$ is generally defined as the change in $m$. If the mass reduces, then $\Delta m$ is negative.

It's a common point of confusion to think that negative quantities must be subtracted. In general, we write $m + \Delta m$ regardless of whether $\Delta m$ is negative or positive.

Ok, but why do they then write $m - \Delta m_l$ for the boxcar? I guess they do it to distinguish it from the increasing term $\Delta m_r$. Should I always write $m + \Delta m$, except when I have both an increasing and a decreasing term? But I don't see how this makes sense. If I indicate a decreasing mass by $-\Delta m$ in one case, it would seem that I always have do it like this. Isn't there a contradiction between the rocket case $m + \Delta m$ and the boxcar case $m +\Delta m_r - \Delta m_l$?

 

[kuruman]

Furthermore, what matters in variable mass problems is the rate of change $\Delta m/\Delta t$. Since $\Delta t$ is always positive, when the system loses mass (as in the case of a rocket) the ratio is negative; if it gains mass (as in the case of a moving container accumulating rainwater) the ratio is positive.

 

[Rick16]

Furthermore, what matters in variable mass problems is the rate of change $\Delta m/\Delta t$. Since $\Delta t$ is always positive, when the system loses mass (as in the case of a rocket) the ratio is negative; if it gains mass (as in the case of a moving container accumulating rainwater) the ratio is positive.

I am sorry, but I don't understand. In the rocket case $\Delta m$ is negative, and they write $+\Delta m$. In the boxcar case $\Delta m_r$ is positive and $\Delta m_l$ is negative, and they write $+\Delta m_r$ and $-\Delta m_l$. In one case a negative quantity is added, and in the other case a positive quantity is added and a negative quantity is subtracted. I don't see how I can reconcile the two cases.

 

[kuruman]

I am sorry, but I don't understand. In the rocket case $\Delta m$ is negative, and they write $+\Delta m$.

Not quite. If you look in the figure, they write $\Delta m =-|\Delta m|.$ This means that $\Delta m$ is negative. That is also explicitly mentioned in the text

I think you are confusing a symbol representing an algebraic quantity with its value. When you write $x+2=0$, you can see that $x$ is equal to $-2$, a negative number. Should you then write $-x+2=0$ because $x$ is negative? Of course not.

That said, I can see how the derivation in your book can be confusing when you have systems that exchange mass and momentum. You may wish to look at this article that tries to address this confusion by following an approach that departs a bit from the standard textbooks.

 

[PeroK]

I am sorry, but I don't understand. In the rocket case $\Delta m$ is negative, and they write $+\Delta m$. In the boxcar case $\Delta m_r$ is positive and $\Delta m_l$ is negative, and they write $+\Delta m_r$ and $-\Delta m_l$. In one case a negative quantity is added, and in the other case a positive quantity is added and a negative quantity is subtracted. I don't see how I can reconcile the two cases.

They can do what they like. As long as it adds up.

 

[Dale]

Should I always write

You can choose whichever convention you like. You just have to be careful to be clear about the convention you use and be careful to identify the convention that is used by an author.

Whatever convention you choose, it is important to be consistent within a single problem. But you are free to use different conventions in different problems. It looks like your book author does that.