I'm reading Spivak's mechanics book and I'm stumped on some of the math in his derivation of an analytic expression for the motion of a rocket in empty space.(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex]\mathbf{v}(t)[/itex] be the rockets velocity, let [itex]\mathbf{q}(t)[/itex] be the velocity at which fuel is ejected from the rocket, and let [itex]m(t)[/itex] be it's mass. Suppose [itex]m'(t) = -k[/itex] for some constant k.

From a stationary frame of reference, the ejection velocity of the fuel is [itex]\mathbf{v}(t) + \mathbf{q}(t)[/itex]. In a small time interval, the amount of fuel ejected is [itex]m(t) - m(t+h)[/itex], and its ejection velocity is approximately [itex]\mathbf{v}(t) + \mathbf{q}(t)[/itex]. Hence the total momentum is

[tex] \mathbf{p}_{fuel}(t) = [m(t) - m(t+h)] \cdot (\mathbf{v}(t) + \mathbf{q}(t))[/tex]

Spivak claims the time derivative at time t of the momentum of the expelled fuel is

[tex] \lim_{h\rightarrow 0} \frac{m(t)-m(t+h)}{h} ( \mathbf{v}(t) + \mathbf{q}(t) = - m'(t) \cdot ( \mathbf{v}(t) + \mathbf{q}(t) ) [/tex]

Here's where I'm confused. I'm not sure how he calculates the time derivative of the momentum of the ejected fuel. I'm assuming he uses the formula

[tex] \lim_{h\rightarrow 0} \frac{\mathbf{p}(t) - \mathbf{p}(t+h)}{h}[/tex]

But when I do it this way, I get that [itex]\mathbf{p}_{fuel}(t) = m'(t) \cdot (\mathbf{v}(t) - \mathbf{p}(t)[/itex] (I'm not even sure if my evaluation of the limit is correct). Perhaps the negative sign denotes the momentum of the ship, but nonetheless, I would really appreciate if someone could explain/show this step to me.

Next Spivak sets his time derivative of momentum equal to the time derivative of [itex] -m(t) \mathbf{v}(t) [/itex]. I'm a bit confused here as well. I know that the time derivative of momentum is equal to the force, and the second law tells us that the sum of the forces is equal to the product of mass and acceleration. So again it looks like he's using the negative sign to denote the velocity of the ship? Again, could someone explain this part to me as well?

At any rate, in case you were curious, Spivak determines that

[tex]m(t) \frac{d \mathbf{v}(t)}{dt} = \frac{dm(t)}{dt} \mathbf{q}(t)[/tex]

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# Rocket in empty space

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