The speed of light theory predicts that as things travel faster their mass increases, so I think we if we look at a plane accelerating from mach one to mach two and measure the relativistic mass of the plane, we'll see an increase in relativistic mass regardless of how small. Is this true?
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The relativistic mass is an obsolete and confusing concept.
The mass of an object is the rest mass.
The $\gamma$ factor in the 4-velocity and hence in the 4-momentum means that to accelerate an object you need more and more energy expenditure and that the speed limit for any object is the speed of light. In principle reachable with an infinite spending of energy.
Michele Grosso
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The real quantity that increases is the relativistic energy, which is given by $E = \gamma mc^2 $ with $\gamma$ being the Lorentz factor $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.
"Relativistic mass" comes from rolling the $\gamma$ into the $m$, such that you have $m' = \gamma m$ and $E=m'c^2$. Whether you do this is mostly a notational choice, since the results are the same either way, but generally it's not done because $\gamma$ shows up a lot in ways that don't look like a change in mass, so it's helpful to keep that seperate.
With that didactic note out of the way, we can estimate what the change in energy/mass for a plane moving from mach 1 to mach 2 would be. Mach 1 is about $10^{-6}\, c$, so the change in the Lorentz factor is about $10^{-12}$, which means that the change in energy for a Jumbo Jet (~100 tons) corresponds to a hundred extra micrograms; a bit less than the energy change if an extra fruit fly stowed on board. This is negligible, so classical mechanics is a good approximation at plane-speeds, as we would expect.
Tetra
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