Why Does F Equal (mdv/dt = dmv/dt)?

[OcaliptusP]

The question i want to ask is why
$$F=\frac{mdv}{dt}=\frac{dmv}{dt}$$
I mean how can we move we in the fraction?
Is there are mathematical proof or comes from something else?

 

[fresh_42]

The question i want to ask is why
$$F=\frac{mdv}{dt}=\frac{dmv}{dt}$$
I mean how can we move we in the fraction?
Is there are mathematical proof or comes from something else?

You can either simply say because $m$ is a constant, i.e. $m$ doesn't vary in time, or you could do it the long way with the Leibniz rule $\frac{d}{dt}(m\cdot v)= (\frac{d}{dt}m)\cdot v + m \cdot \frac{d}{dt}v = 0 \cdot v + \frac{m \cdot dv}{dt}$ which also uses that $m$ is independent of time, such that its differentiation along time becomes zero. So whether one uses $\frac{d}{dt}(c \cdot F(t))= c \cdot \frac{d}{dt}F(f)$ or $\frac{d}{dt}c = 0$ doesn't matter. As long as the mass doesn't vary in time, the two expressions are equal.

 

[OcaliptusP]

Thanks for all your help.

 

[pixel]

There are cases where m is not constant with respect to time, such as in the differential equation for a rocket.