To determine the local speed of sound given the Mach number and the speed of the airplane, follow these steps:
1. Identify the given information:
- Mach number ([tex]\(M\)[/tex]) = 0.81
- Speed of the airplane ([tex]\(v_{airplane}\)[/tex]) = 850 km/h
2. Recall the relationship between Mach number, speed of the airplane, and the speed of sound:
- The Mach number is defined as the ratio of the speed of an object moving through a fluid to the local speed of sound.
- Mathematically, [tex]\( M = \frac{v_{airplane}}{v_{sound}} \)[/tex]
- Rearranging this formula to solve for the local speed of sound ([tex]\( v_{sound} \)[/tex]):
[tex]\[
v_{sound} = \frac{v_{airplane}}{M}
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
v_{sound} = \frac{850 \text{ km/h}}{0.81}
\][/tex]
4. Perform the calculation:
- Divide 850 km/h by 0.81 to determine the local speed of sound.
- This results in:
[tex]\[
v_{sound} \approx 1049.3827160493827 \text{ km/h}
\][/tex]
5. State the final answer:
- Therefore, the local speed of sound is approximately 1049.38 km/h.
Thus, the local speed of sound, given a Mach number of 0.81 and an airplane speed of 850 km/h, is approximately 1049.38 km/h.
