My Attempt At Re-entry Temp. Calculation

I wanted to figure out the temperature(t₂) of any object based on things such as its specific heat capacity, re-entry speed, etc.

Work: W = Fy

Force by Air: F_D = 1/2pv²C_DA

Energy-temperature relationship: Q = mc(t₂ - t₁)

Setting things equal & solving for t₂ gives:

t₂ = (1/2pv²C_DAy+mct₁) / mc

(I'm sorry I could not get latex to format all of the formulas correctly so I formatted none of them...)

So, how accurate is this equation in describing temperature gain through freefall back down to earth?