Proving the Relativistic Energy-Momentum relation, the hard way and the easier way.
This adds the proof for the relativistic mass effects from motion, proving that gravity causes it.
Two reference frames in motions are not symmetrical. A system at rest on the surface of a planet moves faster in time than a moving system. This is not the case using just Special Relativity. This is proven just using the gravitational metric field in General Relativity.
To determine which tank empties first, we need to model the draining process for each based on their shapes. Both tanks start with the same volume (1000 liters) and have identical outlets at the bottom, so we assume the same orifice area and fluid properties. The key physics principle is Torricelli's law: the efflux velocity at the outlet is
v=√2ghv = \sqrt{2gh}
, where (h) is the height of the fluid surface above the outlet, and (g) is gravity. The flow rate out is
Q=a√2ghQ = a \sqrt{2gh}
, where (a) is the effective outlet area (assumed constant and the same for both).The time to empty is found by considering how the volume changes:
dV=-A(h)dhdV = -A(h) dh
(where (A(h)) is the cross-sectional area at height (h)), leading to
dh/dt=-(a/A(h))√2ghdh/dt = - (a / A(h)) \sqrt{2gh}
. Integrating gives the total time
T=1/(a√2g) ∫_0^H▒(A(h))/√h dhT = \frac{1}{a \sqrt{2g}} \int_0^H \frac{A(h)}{\sqrt{h}} \, dh
, where (H) is the initial height. Since the constant
1/(a√2g)\frac{1}{a \sqrt{2g}}
is the ...
