The equation \( F = \frac{mdv}{dt} = \frac{dmv}{dt} \) holds true under the condition that mass \( m \) remains constant over time. When mass is constant, the differentiation of mass with respect to time is zero, allowing the terms to be manipulated as shown. The Leibniz rule confirms this by demonstrating that \( \frac{d}{dt}(m \cdot v) \) simplifies to \( m \cdot \frac{dv}{dt} \) when \( m \) is independent of time. In scenarios where mass varies, such as in rocket dynamics, this equality does not apply.
PREREQUISITESPhysics students, engineers, and anyone interested in the mathematical foundations of motion and force, particularly in contexts involving variable mass systems.
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 said: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?