- #1

Hak

- 709

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Speaking of variable mass systems... it seems to me that there is a flaw in Halliday's reasoning when he talks about this subject. But the formula he derives seems to work to me!

He starts with a system of points of total mass [tex]\ M[/tex] and cm velocity [tex]\ v[/tex]. Due to the action of external forces after a certain time t the system has clearly split into two parts. The first has mass [tex]\ M -\Delta M[/tex] and cm velocity [tex]\ v +\Delta v[/tex]; the second has mass [tex]\Delta M[/tex] and velocity [tex]\ u[/tex].

Applying [tex]F_{est} = \frac{\Delta P}{\Delta t}[/tex] finds that for a finite interval of time it holds (approximately)

[tex]\ F_{est} = M \frac{\Delta v}{\Delta t} + [u-(v+\Delta v)]\frac{\Delta M}{\Delta t}[/tex].

Moving on to the limit for t tending to 0

replaces [tex]\frac{\Delta v}{\Delta t}[/tex] with [tex]\frac{dv}{dt}[/tex]

replaces [tex]\frac{\Delta M}{\Delta t}[/tex] with [tex]-\frac{dM}{dt}[/tex]

places [tex]\Delta v = 0[/tex].

Thus [tex]\ F_{est} = M \frac{dv}{dt} + v \frac{dM}{dt} -u \frac{dM}{dt}[/tex].

But shouldn't it have taken into account that [tex]\ u[/tex] also goes to zero (or at least changes) for t that tends to zero?

Am I wrong in making such an assumption or am I right? Thanks in advance.

He starts with a system of points of total mass [tex]\ M[/tex] and cm velocity [tex]\ v[/tex]. Due to the action of external forces after a certain time t the system has clearly split into two parts. The first has mass [tex]\ M -\Delta M[/tex] and cm velocity [tex]\ v +\Delta v[/tex]; the second has mass [tex]\Delta M[/tex] and velocity [tex]\ u[/tex].

Applying [tex]F_{est} = \frac{\Delta P}{\Delta t}[/tex] finds that for a finite interval of time it holds (approximately)

[tex]\ F_{est} = M \frac{\Delta v}{\Delta t} + [u-(v+\Delta v)]\frac{\Delta M}{\Delta t}[/tex].

Moving on to the limit for t tending to 0

replaces [tex]\frac{\Delta v}{\Delta t}[/tex] with [tex]\frac{dv}{dt}[/tex]

replaces [tex]\frac{\Delta M}{\Delta t}[/tex] with [tex]-\frac{dM}{dt}[/tex]

places [tex]\Delta v = 0[/tex].

Thus [tex]\ F_{est} = M \frac{dv}{dt} + v \frac{dM}{dt} -u \frac{dM}{dt}[/tex].

But shouldn't it have taken into account that [tex]\ u[/tex] also goes to zero (or at least changes) for t that tends to zero?

Am I wrong in making such an assumption or am I right? Thanks in advance.

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