Question

An aeroplane is flying at a height of 15 km, where the temperature is -50°C. Assuming \( k = 1.4 \) and \( R = 287 \, \text{J/K . kg} \), the approximate speed of the plane corresponding to \( M = 2.0 \) will be?

• A. 1955 km/hour
• B. 2055 km/hour
• C. 2155 km/hour
• D. 2255 km/hour

Hint:

To calculate the speed corresponding to a given Mach number, use the formula \( v = M \times c \), where \( c \) is the speed of sound calculated using \( c = \sqrt{k R T} \).

Answer 1

The Correct Option is B

Step 1: Use the formula for Mach number. 
The Mach number \( M \) is given by: \[ M = \frac{v}{c} \] Where \( v \) is the speed of the plane, and \( c \) is the speed of sound, which is calculated by: \[ c = \sqrt{k R T} \] Here, \( T = -50^\circ \text{C} = 223.15 \, \text{K} \), and \( k = 1.4 \), \( R = 287 \, \text{J/K . kg} \). \[ c = \sqrt{1.4 \times 287 \times 223.15} \approx 340.29 \, \text{m/s} \] Step 2: Calculate the speed of the plane. 
Given \( M = 2.0 \), the speed \( v \) is: \[ v = M \times c = 2.0 \times 340.29 \approx 680.58 \, \text{m/s} = 2055 \, \text{km/hour} \] Final Answer: \[ \boxed{2055 \, \text{km/hour}} \]