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Consider the one-dimensional flow of quantity $q(x,t)$. The advection equation is generalized to, $$\partial_tq+\partial_xF(q,x)=0\tag{1}$$ where $F(q,x)$ is the flux of $q$ in the flow--in the case of the advection equation $F(q,x)=qu$. However, we can use the chain rule to write the above as  \partial_tq+\frac{\partial F}{\partial q}\frac{\partial ...

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The number of particles is a conserved quantity (at large). The mass unit is derived from the 'number of particles'. Take the mass unit, the kilogram as a 'bunch of atoms' . The atomic unit is a fraction of a 'bunch of atoms'. It is a recursive definition. Which means that, as all the 'main' equations of physics are reported to the 'electron mass', the ...

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Other answers and the referenced paper assume a constant radius turn. This path would require a discontinuous steering angle, which is not only non-physical but a somewhat poor approximation for how human drivers drive. The reason drivers don't approximate this technique (besides that it would take very rapid steering wheel movement) is that it would incite ...

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The thrust coming from a rocket engine is exerted on the engine bell, and it is directed along its axis of symmetry. It's not completely clear how you're modelling your ship but it is probably more realistic to apply the force to the "thruster fire" block, whatever that is. It's important to note, though, that if applying the force to the engine bell and ...

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If you plot the path of the moon with respect to the star you should get a cycloid pattern averaging around the path of the planet. That's what our moon does. The planet and the moon are doing a gyrating dance about their center of mass, with the moon doing the most motion (because it has the smallest mass), while the planet/moon duo orbit the star. I've ...

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The issue here is that your front wheels are turned/steered by the same angle. When you try to find the instantaneous centre of curvature, you may first want to assume the wheels won't slip from side to side, like you may get if you drive around a corner on a slippy road. As there is no slip, the velocity of each wheel must occur in the direction the ...

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N-body simulations in full general relativity are difficult because gravity is a field theory and because it is non-linear. Let's deal with the field theory part first. In Newtonian mechanics gravity is static. The field itself has no energy or momentum, no degrees of freedom at all. It is simply an instantaneous force law between all matter. Remove the ...

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