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aghostinthefigures
aghostinthefigures
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This is a subtle little problem that has to do with what Bernoulli's equation actually states; that there exists a relationship between the pressure $p$ and the velocity $\vec{v}$ such that, in the absence of external work/forces, $$p + \frac{\rho}{2}\vec{v}\cdot\vec{v} = C$$ where $C$ is a constant, representing a sort of energy density, and $\rho$ is the density.

You have correctly spotted that the $\frac{1}{2}\vec{v}\cdot\vec{v}$$\frac{\rho}{2}\vec{v}\cdot\vec{v}$ term is not Galilean-invariant. The challenge is that, neither is $C$! You can't perform a change of reference frame and use the old value of $C$ and the transformed value of $\vec{v}$ to calculate the pressures in the new frame of your system; you need additional information about how your system is set up in order to solve for the pressures, keeping in mind that external work is also not Galilean-invariant.

This is a subtle little problem that has to do with what Bernoulli's equation actually states; that there exists a relationship between the pressure $p$ and the velocity $\vec{v}$ such that, in the absence of external work/forces, $$p + \frac{\rho}{2}\vec{v}\cdot\vec{v} = C$$ where $C$ is a constant, representing a sort of energy density, and $\rho$ is the density.

You have correctly spotted that the $\frac{1}{2}\vec{v}\cdot\vec{v}$ term is not Galilean-invariant. The challenge is that, neither is $C$! You can't perform a change of reference frame and use the old value of $C$ and the transformed value of $\vec{v}$ to calculate the pressures in the new frame of your system; you need additional information about how your system is set up in order to solve for the pressures, keeping in mind that external work is also not Galilean-invariant.

This is a subtle little problem that has to do with what Bernoulli's equation actually states; that there exists a relationship between the pressure $p$ and the velocity $\vec{v}$ such that, in the absence of external work/forces, $$p + \frac{\rho}{2}\vec{v}\cdot\vec{v} = C$$ where $C$ is a constant, representing a sort of energy density, and $\rho$ is the density.

You have correctly spotted that the $\frac{\rho}{2}\vec{v}\cdot\vec{v}$ term is not Galilean-invariant. The challenge is that, neither is $C$! You can't perform a change of reference frame and use the old value of $C$ and the transformed value of $\vec{v}$ to calculate the pressures in the new frame of your system; you need additional information about how your system is set up in order to solve for the pressures, keeping in mind that external work is also not Galilean-invariant.

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aghostinthefigures
aghostinthefigures
  • 3.3k
  • 11
  • 31

This is a subtle little problem that has to do with what Bernoulli's equation actually states; that there exists a relationship between the pressure $p$ and the velocity $\vec{v}$ such that, in the absence of external work/forces, $$p + \frac{\rho}{2}\vec{v}\cdot\vec{v} = C$$ where $C$ is a constant, representing a sort of energy density, and $\rho$ is the density.

You have correctly spotted that the $\frac{1}{2}\vec{v}\cdot\vec{v}$ term is not Galilean-invariant. The challenge is that, neither is $C$! You can't perform a change of reference frame and use the old value of $C$ and the transformed value of $\vec{v}$ to calculate the pressures in the new frame of your system; you need additional information about how your system is set up in order to solve for the pressures, keeping in mind that external work is also not Galilean-invariant.