So in my lectures we derived the rocket equation which is given by:
$$F = m \cdot\frac{dv}{dt} + u\cdot \frac{dm}{dt}$$
Where m is the mass of the rocket, and u is the speed of the ejected fuel.
But if I use
$$F = \frac{dP}{dt} = \frac{d(mv)}{dt}$$
Where $P$ is the momentum I get
$$F = m \cdot\frac{dv}{dt} + v\cdot \frac{dm}{dt}$$
My question is why are these different? Isn't this a contradiction?
