In compressible-flow approximations, gases may be treated as incompressible when the Mach number (M) is approximately:

Introduction / Context:
Compressibility effects in gases become significant as flow speed approaches the speed of sound. For many engineering calculations, a threshold Mach number is used to justify the incompressible assumption and the use of simpler formulas.

Given Data / Assumptions:

• Mach number M = V/a, where a is the local speed of sound.
• Small density change criterion for incompressible modelling.
• Low subsonic regime.

Concept / Approach:
Empirical and theoretical analyses show that when M < 0.3, compressibility effects are minor for many applications; a more conservative and commonly taught rule-of-thumb is M < 0.2, under which density change is negligible and incompressible flow theory applies with good accuracy.

Step-by-Step Solution:
Check regime: M < 0.2 → negligible compressibility.Therefore, continuity and momentum equations can be used with constant density.

Verification / Alternative check:
Isentropic relations show density change scales with M^2 for small M; at M = 0.2, changes are within a few percent, acceptable for many designs.

Why Other Options Are Wrong:
M ≥ 0.5: compressibility influential; shocks not present but density change is non-negligible.M ≈ 1 or 1.5: transonic/supersonic—strongly compressible.

Common Pitfalls:

• Applying incompressible Bernoulli at high Mach numbers.
• Ignoring temperature variations tied to compressibility.

Final Answer:
Less than 0.2