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I would like to know what speed would be required, at sea level, to exert the same aerodynamic pressure as experienced at cruise altitude.

For example, if an airliner were traveling at 500kn at 35,000' what speed would you travel at sea level to experience the same forces on the air frame?

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You are looking for the equivalent airspeed. Here is the definition from Wikipedia:

Equivalent airspeed (EAS) is calibrated airspeed (CAS) corrected for the compressibility of air at a non-trivial Mach number. It is also the airspeed at sea level in the International Standard Atmosphere at which the dynamic pressure is the same as the dynamic pressure at the true airspeed (TAS) and altitude at which the aircraft is flying.

$$ \text{EAS} = \text{TAS} \times \sqrt{\frac{\text{density at altitude}}{\text{density at standard sea level}}} $$

At standard sea level, EAS is the same as calibrated airspeed (CAS) and true airspeed (TAS). At any other altitude, EAS may be obtained from CAS by correcting for compressibility error.

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Assuming you mean 500 kts True Airspeed, it's whatever the indicated/calibrated airspeed shows (what the pilot sees on the airspeed tape), because that's based on the actual dynamic pressure at altitude. Using any TAS calculator to work it backwards, a true airspeed of 500 kts is 243 kts indicated on a standard day (15C and 29.92" at sea level), so the dynamic pressure the airplane "feels" at 35000 feet is equivalent 243 kts True Airspeed at sea level.

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