Determine the Mach number of a car moving in standard air
at a speed of (a) 25 mph, (b) 55 mph, and (c) 100 mph.
The Correct Answer and Explanation is :
To determine the Mach number of a car moving at different speeds, we use the formula: M=vaM = \frac{v}{a}
Where:
- MM is the Mach number,
- vv is the speed of the vehicle,
- aa is the speed of sound in air.
Step 1: Determine the Speed of Sound in Air
The speed of sound in air depends on the temperature and the pressure, but in standard conditions (sea level, 15°C), the speed of sound in dry air is approximately 343 meters per second (m/s).
Step 2: Convert the Car Speeds to Meters per Second
We are given the speeds of the car in miles per hour (mph). To use the formula, we need to convert these values to meters per second (m/s). The conversion factor is: 1 mph=0.44704 m/s1 \, \text{mph} = 0.44704 \, \text{m/s}
Now, let’s convert each speed:
- For 25 mph: 25 mph=25×0.44704=11.176 m/s25 \, \text{mph} = 25 \times 0.44704 = 11.176 \, \text{m/s}
- For 55 mph: 55 mph=55×0.44704=24.587 m/s55 \, \text{mph} = 55 \times 0.44704 = 24.587 \, \text{m/s}
- For 100 mph: 100 mph=100×0.44704=44.704 m/s100 \, \text{mph} = 100 \times 0.44704 = 44.704 \, \text{m/s}
Step 3: Calculate the Mach Number
Now that we have the speeds in meters per second, we can calculate the Mach number for each speed using the formula M=vaM = \frac{v}{a}.
- For 25 mph (11.176 m/s): M=11.176343≈0.0326M = \frac{11.176}{343} \approx 0.0326
- For 55 mph (24.587 m/s): M=24.587343≈0.0717M = \frac{24.587}{343} \approx 0.0717
- For 100 mph (44.704 m/s): M=44.704343≈0.1304M = \frac{44.704}{343} \approx 0.1304
Conclusion:
- At 25 mph, the car is moving at a Mach number of approximately 0.033.
- At 55 mph, the car is moving at a Mach number of approximately 0.072.
- At 100 mph, the car is moving at a Mach number of approximately 0.130.
Explanation:
The Mach number represents the ratio of the speed of an object to the speed of sound. When the Mach number is less than 1, it indicates subsonic speed (the object is moving slower than the speed of sound). These Mach numbers are far below 1, meaning the car is moving at subsonic speeds at all these speeds. Generally, speeds higher than Mach 1 would indicate supersonic motion, which is significantly faster than the speeds of these cars.