The generalization of Newton's 2nd Law to apply to variable mass systems

[sedaw]

F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt) = a + v(dm/dt)

i don't understand why d(mv)/dt = m(dv/dt) + v(dm/dt)
can someone help ?

TNX !

 

[ehild]

How do you calculate the derivative of a product?

ehild

 

[EngineerHead]

This is simply obtained by using the product rule.

dp/dt = d/dt (mv) = m(dv/dt) + v(dm/dt) (using product rule).

 

[Jokerhelper]

I thought that Newton's second law was true only for systems with constant mass.

 

[irycio]

Well, written in the form of
$$F=m \frac{dv}{dt}$$
it is valid inly for constant mass system since you assume that the mass is constant :P.

But assuming, that mass is dependent on velocity, which is true (according to SR) you get what sedaw wrote.
Thus, the best, most general form of Newton's 2nd law is
$$F=\frac{dp}{dt}$$
as it doesn't imply anything being constant ;P

 

[Jokerhelper]

Well, written in the form of
$$F=m \frac{dv}{dt}$$
it is valid inly for constant mass system since you assume that the mass is constant :P.

But assuming, that mass is dependent on velocity, which is true (according to SR) you get what sedaw wrote.
Thus, the best, most general form of Newton's 2nd law is
$$F=\frac{dp}{dt}$$
as it doesn't imply anything being constant ;P

I know that $$\bold{F}=\frac{d\bold{p}}{dt}$$ is Newton's second law, but I read that this was true only with constant masses. Also, I don't know much about special relativity (I just finished my freshman year) but I thought it proved that Galilean transformation was flawed. Since Newton's laws are based on Galilean relativity, I don't think SR can show that N2 holds even for variable masses. Can it?

 

[diazona]

Well, what you read was wrong. In nonrelativistic physics, $\mathbf{F} = \mathrm{d}\mathbf{p}/\mathrm{d}t$ is valid regardless of whether mass changes.

If I remember correctly, in special relativity, Newton's second law (in the above form) is used to define the relativistic generalization of force. So that equation holds true in all cases. (Caveat: there are actually a couple of ways to define the relativistic generalization of force that are not quite compatible with each other... it turns out that force is not as useful a concept in special relativity as it is in non-relativistic, Newtonian physics.)