Introduction / Context:
In aerodynamics and flight mechanics, the speed of sound and the Mach number are foundational concepts. This question checks understanding of (1) the numerical value of the speed of sound at standard sea-level conditions, (2) how it varies with temperature, and (3) what Mach number means with respect to local conditions.
Given Data / Assumptions:
- Standard sea-level temperature T0 = 0°C for the stated comparisons.
- Speed of sound at 0°C is approximately 331 m/s (about 1190–1192 km/h).
- Mach number M is defined as V / a, where a is the local speed of sound.
- Aircraft speed example: 2975 km/h is compared with Mach at 0°C.
Concept / Approach:
The speed of sound in air varies with absolute temperature as a = sqrt(gamma * R * T). To first order near sea level, a increases roughly 0.6 m/s per °C. Mach 1 is, by definition, the local speed of sound at the existing temperature and pressure; “2.5 Mach” means 2.5 times the local speed of sound.
Step-by-Step Solution:
1) Statement 1: “Speed of sound at 0°C is 1190 km/h.” Using 331 m/s * 3.6 = 1191.6 km/h, this is essentially correct within rounding.2) Statement 2: “Sound varies 2.4 km/h per degree rise.” Using 0.6 m/s per °C * 3.6 = 2.16 km/h per °C, so 2.4 km/h per °C overstates the change. This is inaccurate.3) Statement 3: “Sound at 0°C is called one Mach.” Interpreted as “Mach 1 equals the local speed of sound at 0°C,” this aligns with the definition of Mach (V/a). Wording is awkward but substantively correct.4) Statement 4: “An aircraft flying at 2975 km/h is called 2.5 Mach.” Using 1190 km/h (0°C), 2.5 * 1190 = 2975 km/h. This is correct under the stated reference.
Verification / Alternative check:
Compute 331 m/s * 3.6 = 1191.6 km/h (confirms Statement 1). 2.5 * 1190 ≈ 2975 (confirms Statement 4).
Why Other Options Are Wrong:
Options that include Statement 2 as correct are wrong because 2.4 km/h per °C is not the commonly accepted slope; ~2.16 km/h per °C is more accurate.“All statements correct” is wrong because Statement 2 is not correct.
Common Pitfalls:
Confusing exact vs approximate values; mixing m/s per °C with km/h per °C; treating Mach number as fixed irrespective of local temperature.
Final Answer:
Only statements 1, 3, and 4 are correct.
Discussion & Comments