Question

practice

1. if the local speed of sound is 344 m/s and an aircraft is flying at 910 km/h, what the mach number? ans: 0.73
2. if the mach number is 0.93 and the local speed of sound is 320 m/s, what is th speed of an airplane in these conditions? ans: 3.0 × 10² m/s = 1100 km/h
3. if the mach number is 0.81 and the speed of an airplane measured by radar is 850 km/h, what is the local speed of sound in kilometres per hour?

ans: 290 m/s = 1.0 × 10³ km/h

Explanation:

Problem 1:

Step1: Convert aircraft speed to m/s

First, we need to convert the aircraft speed from km/h to m/s. We know that \( 1 \text{ km} = 1000 \text{ m} \) and \( 1 \text{ h} = 3600 \text{ s} \). So, the conversion factor is \( \frac{1000}{3600}=\frac{5}{18} \text{ m/s per km/h} \).
The aircraft speed is \( 910 \text{ km/h} \), so in m/s it is \( 910\times\frac{5}{18}\approx252.78 \text{ m/s} \)

Step2: Calculate Mach number

The Mach number \( M \) is given by the formula \( M=\frac{v}{c} \), where \( v \) is the speed of the aircraft and \( c \) is the speed of sound.
Here, \( v = 252.78 \text{ m/s} \) and \( c = 344 \text{ m/s} \)
So, \( M=\frac{252.78}{344}\approx0.73 \)

Step1: Use Mach number formula

The formula for Mach number is \( M = \frac{v}{c} \), where \( M \) is Mach number, \( v \) is aircraft speed and \( c \) is speed of sound. We need to find \( v \), so we can rearrange the formula as \( v = M\times c \)

Step2: Substitute values

Given \( M = 0.93 \) and \( c = 320 \text{ m/s} \)
So, \( v=0.93\times320 = 297.6\approx3.0\times 10^{2} \text{ m/s} \)
To convert to km/h, we use the conversion factor \( \frac{3600}{1000} = 3.6 \text{ km/h per m/s} \)
So, \( 297.6\times3.6 = 1071.36\approx1100 \text{ km/h} \)

Step1: Use Mach number formula to find speed of sound in km/h

The Mach number formula is \( M=\frac{v}{c} \), where \( M = 0.81 \), \( v = 850 \text{ km/h} \) and we need to find \( c \) (speed of sound in km/h). Rearranging the formula gives \( c=\frac{v}{M} \)

Step2: Substitute values

Substituting \( v = 850 \text{ km/h} \) and \( M = 0.81 \)
\( c=\frac{850}{0.81}\approx1049.38\approx1.0\times 10^{3} \text{ km/h} \)
To convert to m/s, we use the conversion factor \( \frac{1000}{3600}=\frac{5}{18} \text{ m/s per km/h} \)
\( 1049.38\times\frac{5}{18}\approx291.49\approx290 \text{ m/s} \)

Answer:

\( 0.73 \)

Problem 2: