High velocity mushrooms? So y’all know meteoroites burn up in our atmosphere I’m looking for an equation or something that will allow me to calculate how fast a substance can move before it starts to burn up. Specifically I’m asking because I want to calculate the maximum velocity a mushroom can reach before it becomes a roasted mushroom. 
 A: XKCD's "What If?" has a related scenario worth reading, Steak drop: From what height would you need to drop a steak for it to be cooked when it hit the ground?
As noted there, getting a singed surface is not too hard, but cooking is trickier since the time spent surrounded by hot air is not long. In the mushroom case the mass is lower and the surface area higher (plus, to cook mushrooms you need higher temperature IMHO than for a steak), so the effect might be even less. 
The essay refers to this page/calculator of the stagnation temperature, giving formulas that are not terribly neat but soluble. 
If we use the simpler ideal formula $$T_{stagnation}= T_{static} \left[1 + M^2 \frac{\gamma-1}{2}\right]$$ (where $T_{static}$ is the temperature at that level of the atmosphere, $M$ the Mach number $v/c_s$ where $C_s$ is the speed of sound, and $\gamma=1.4$ for a perfect gas), then for a burning temperature $T_{burn}$ we can just invert it and get that $$v_{burn} = c_s \sqrt{\frac{2}{\gamma-1}\left(\frac{T_{burn}}{T_{static}}-1\right)}.$$ For at atmosphere at 300 K with $c_s=343$ m/s and $T_{burn}=477$ K (this corresponds to oven roasting of vegetables) I get 589.1218 m/s. 
But this is assuming a lot about proper heat transfer: I suspect that will be imperfect (as discussed in the what if) and the necessary speed is larger. It is also worth noting that this is far above terminal velocity so unless a force is accelerating the mushroom continually it will not heat up much. 
Happy cooking!
