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So, this is a question that just came to me. We all know that the derivative of momentum with respect to time is the definition of a force. $$ \frac{{\rm d}p}{{\rm d}t}=\frac{{\rm d}mv}{{\rm d}t}=m\frac{{\rm d}v}{{\rm d}t}=ma $$ However, that's assuming that the mass remains constant. What if the mass was changing? Such as if we had a rocket that lost like 90% of its mass as fuel? Well then, wouldn't the mass also be a variable there, so you wouldn't be able to just factor it out and treat it as a constant.

So how exactly would you find the force applied then?

UPDATE: I took calculus and understand why this question is pretty stupid

Qmechanic
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Physicyquestionasker
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2 Answers2

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You would simply just use the product rule. The definition of net force $\mathbf{F}$ is always given by

$$ \sum_i \mathbf{F}_i = \frac{d\mathbf{p}}{d t} = \frac{d}{dt} m(t)\mathbf{v}=\mathbf{v}\frac{dm}{dt} + m(t)\frac{d\mathbf{v}}{dt}.$$

InertialObserver
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We can do this directly by using the rate of change of the mass, $\mathrm{d}m/\mathrm{d}t$, such as the rate of fuel consumption. Then use $F = v(\mathrm{d}m/\mathrm{d}t)$ and you will get the answer.

Vincent Thacker
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user430781
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