“The density of air is the same at different heights (altitudes).” Evaluate the statement.

Introduction:
Atmospheric density varies with altitude due to changes in pressure and temperature. Recognizing this variation is essential in aerodynamics, meteorology, and high-altitude engineering.

Given Data / Assumptions:

• Standard atmosphere model (pressure decreases with height).
• Air treated as a compressible gas.

Concept / Approach:
Hydrostatic balance shows pressure decreases with height. Since density rho is related to pressure p and temperature T via the ideal gas relation rho = p / (R * T), changes in p and T with altitude cause rho to change as well. Therefore, density is not the same at different heights.

Step-by-Step Solution:
1) Recognize dp/dz = -rho * g (hydrostatic relation).2) As z increases, p decreases, reducing rho unless temperature rises enough to compensate (which it usually does not).3) Hence, density varies with altitude; it decreases significantly in the troposphere and beyond.

Verification / Alternative check:
Standard atmosphere tables show sea-level density about 1.225 kg/m^3 dropping to roughly 0.736 kg/m^3 at 2500 m and far lower at high altitudes, confirming non-uniformity.

Why Other Options Are Wrong:

• Correct: Contradicts well-established atmospheric physics.
• True only up to a few meters: Even tens of meters show measurable though small changes; the statement remains inaccurate.
• True in isothermal layers: Even if temperature is constant, pressure still falls with height so density changes.

Common Pitfalls:

• Assuming air is incompressible in large-scale atmospheric problems.
• Using sea-level density for high-altitude performance estimates without correction.

Final Answer:
Incorrect