Problem 58 - Master Resource Book in JEE Main Physics

Question

The concorde is the fastest airlines used for commercial service. It can cruise at 1450 mile per hour (about two times the speed of sound or in other words mach 2). What is it in \(\mathrm{m} / \mathrm{s}\) ? (a) \(644.4 \mathrm{~m} / \mathrm{s}\) (b) \(80 \mathrm{~m} / \mathrm{s}\) (c) \(40 \mathrm{~m} / \mathrm{s}\) (d) None of these

Step-by-Step Solution

Answer

The speed in meters per second is approximately 644.4 \(\text{m/s}\) (option a).

Step 1: Identify the Given Information

We know that the Concorde's speed is given as 1450 miles per hour. Our task is to convert this speed into meters per second (\(\text{m/s}\)).

Step 2: Convert Miles to Kilometers

We need to use the conversion factor where 1 mile is approximately equal to 1.60934 kilometers. Thus, convert 1450 miles as follows:\[1450 \text{ miles} \times 1.60934 \frac{\text{km}}{\text{mile}} = 2334.553 \text{ km}\]

Step 3: Convert Kilometers to Meters

Since there are 1000 meters in a kilometer, we need to convert the speed from kilometers to meters.\[2334.553 \text{ km} = 2334.553 \times 1000 = 2334553 \text{ meters}\]

Step 4: Convert Hours to Seconds

Now, we need to convert the time unit from hours to seconds. Since there are 3600 seconds in an hour, we can divide the total meters by 3600 to find the speed in meters per second:\[\frac{2334553 \text{ meters}}{3600 \text{ seconds}} \approx 648.48 \text{ m/s}\]

Step 5: Compare with Provided Options

The calculated speed is approximately 648.48 \(\text{m/s}\). The closest option to this value is (a) 644.4 \(\text{m/s}\), which seems to be an approximate rounded value.

Key Concepts

Speed of Sound

The speed of sound is a fundamental concept in physics and engineering. It refers to how fast sound waves can travel through a medium, such as air, water, or solids. In air, under standard conditions (at sea level and at 20 degrees Celsius), the speed of sound is approximately 343 meters per second (m/s). This speed can change based on conditions such as temperature and air pressure.
Sound waves move quicker in warmer air or denser materials like water or steel.
Understanding the speed of sound is crucial when discussing high-speed travel, especially with aircraft like the Concorde. The speed of sound serves as a natural benchmark. That's why terms like "Mach number" are often used, which relate directly to how many times the speed of sound an object is moving.

Conversion Factors

Conversion factors are numbers used to change the units of measurement without changing the value itself. They are essential for translating one measurement into another. For example, converting speed from miles per hour (mph) to meters per second (m/s) requires using a conversion factor. When working with unit conversions:

• Remember that 1 mile equals approximately 1.60934 kilometers.
• Remember that 1 kilometer equals 1000 meters.
• Also, there are 3600 seconds in one hour.

This combination helps in performing step-by-step transformations from one set of units to another easily. In our example, the Concorde's speed changed from miles per hour to meters per second using these factors. Each multiplication or division was like a small bridge getting us closer to the final converted speed value.

Concorde Aircraft

The Concorde was a groundbreaking, supersonic passenger airliner that could fly faster than the speed of sound. It was jointly developed by the British and French and remained in service from 1976 to 2003. The Concorde's main feature was its ability to cruise at speeds greater than Mach 1 (speed of sound), reaching speeds around Mach 2—twice the speed of sound or approximately 1450 miles per hour.
This aircraft allowed passengers to travel across the Atlantic in about half the time of conventional aircraft. The Concorde was not just about speed. Its design, with a sleek delta-wing and long pointed nose, made it an aviation icon.
With only 20 ever built, it remains one of the great symbols of 20th-century engineering and aviation.

Mach Number

The Mach number is a unitless measure indicating how fast an object is traveling compared to the speed of sound. It serves as a critical reference point in aerodynamics and fluid mechanics.To calculate the Mach number, divide the object's speed by the speed of sound:\[\text{Mach number} = \frac{\text{object speed}}{\text{speed of sound}}\]For the Concorde, cruising at Mach 2 meant it could travel at twice the speed of sound. A high Mach number can influence various factors like aerodynamics and airflow, essential aspects for aircraft design and performance.
High Mach numbers, especially above Mach 1, introduce unique challenges such as shock waves and increased aerodynamic heating, which engineers must address to ensure aircraft performance and safety.