Just a quick addendum to Aviast's otherwise excellent post:
1/2 rho V^2 has rho decreasing with altitude and V = TAS
1/2 rho0 Vi^2 assumes constant sea-level density and indicated airspeed (Vi), where rho0 = 1.225 kg/m^3)
One reason ASIs use a constant density (ISA sea level density) is that density cannot easily be measured and is also quite difficult to calculate, especially when using a mechanical instrument. Also, it has the handy side effect that for some aircraft (i.e. those that do not experience compressibility to any significant extent), scheduled speeds (e.g. Vs, Va, Vne, etc) are not density dependent and thus not temperature dependent or altitude dependent. Imagine how much harder it would be to operate an aircraft where all the safety-related speeds changed with temperature and altitude!
Note to the more technically advanced readers of the forum:
I think I am correct in stating that for a given weight and configuration, an aircraft will stall at the same Equivalent Airspeed at any altitude attainable within the flight envelope, but I didn't want to confuse our young questioner with discussions of IAS/CAS/EAS, pressure error corrections and scale altitude corrections! Anyway, for most light aircraft Vi is roughly Ve given the likely instrument errors and residual pressure errors, and provided the mach number is low enough then the Bernoulli's equation for non-compressible flow applies and thus IAS and TAS can be related simply by root sigma. However, I hope the university teaches its students about Bernoulli's equation for compressible flow and all the additional complexity that comes with it!
