What is the Bernoulli Equation?

AI Thread Summary
The discussion centers around deriving the Bernoulli Equation from a given stagnation ratio equation. The equation presented is Po/P = [1+((k-1)/2)*Ma^2]^(k/(k-1)), where Po represents stagnation pressure, P is static pressure, and V is velocity. The value of k is specified as 1.4, applicable for air, and Ma refers to the Mach number. The poster encourages engagement with the mathematical exercise rather than seeking a solution. The conversation highlights the enjoyment of deriving equations in fluid dynamics.
PhysicsGnome
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Homework Statement


I recently was given an interesting problem I thought I'd share with the community here. It's a fun exercise in mathematics. Given the equation:

Po/P = [1+((k-1)/2)*Ma^2]^(k/(k-1))

From this stagnation ratio, derive the Bernoulli Equation:

Po = P + 1/2 * rho * V^2

Anyway, it's not for homework, so no real need to solve it. Hope you enjoy deriving equations as much as I do. Have fun!
 
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what are k, M, a?
 
k in this case is 1.4 (for air)

Ma is the mach number.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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