Problem 105 - Physics

Question

Concept Question Suppose you are part of a team that is trying to break the sound barrier with a jet-powered car, which means that it must travel faster than the speed of sound in air. Would you attempt this feat early in the morning when the temperature is cool, later in the afternoon when the temperature is warmer, or does it even matter what the temperature is? Problem In the morning, the air temperature is \(0^{\circ} \mathrm{C}\) and the speed of sound is \(331 \mathrm{~m} / \mathrm{s}\). What must be the speed of your car if it is to break the sound barrier when the temperature has risen to \(43^{\circ} \mathrm{C}\) in the afternoon? Assume that air behaves like an ideal gas.

Step-by-Step Solution

Answer

The car must travel faster than 356.22 m/s at 43°C to break the sound barrier.

Step 1: Understanding the Speed of Sound

The speed of sound in air varies with the temperature. At a temperature of \(0^{\circ} \mathrm{C}\), the speed of sound is known to be \(331 \mathrm{~m/s}\). As the temperature increases, the speed of sound increases as well.

Step 2: Use the Speed of Sound Formula

The formula that relates the speed of sound (\(v\)) with temperature \(T\) in Celsius is given by: \[ v = v_0 \sqrt{\frac{T + 273.15}{T_0 + 273.15}} \] where \(v_0\) is the speed of sound at a reference temperature \(T_0\), which is \(0^{\circ} \mathrm{C}\) in this case.

Step 3: Calculate the New Speed of Sound

Plugging in the values into the formula: \[ v = 331 \sqrt{\frac{43 + 273.15}{0 + 273.15}} \approx 331 \sqrt{\frac{316.15}{273.15}} \] Calculate the square root and then the speed of sound.

Step 4: Solving the Calculation

Calculate the value inside the square root first: \[ \frac{316.15}{273.15} \approx 1.1575 \] Now calculate the square root: \[ \sqrt{1.1575} \approx 1.0764 \] Thus, the new speed of sound is \[ v \approx 331 \times 1.0764 \approx 356.22 \mathrm{~m/s} \]

Step 5: Conclusion

The speed of the car must be greater than \(356.22 \mathrm{~m/s}\) to break the sound barrier when the temperature is \(43^{\circ} \mathrm{C}\). The temperature affects the speed at which sound travels, therefore the car's speed must adjust accordingly.

Key Concepts

Temperature effect on speed of sound

The speed of sound is not a constant value; it changes with the temperature of the medium through which it travels, such as air.
When the temperature increases, the particles in the air move more rapidly.
This increased motion allows sound waves to travel more quickly.

• At 0°C, the speed of sound is approximately 331 m/s.
• When the temperature is higher, for example at 43°C, the speed increases.

This relationship is important when planning to break the sound barrier with a jet-powered car.
If the air temperature is warmer in the afternoon, the car must travel faster to surpass the increased speed of sound.

Jet-powered car

A jet-powered car differs from traditional vehicles as it uses jet engines for propulsion.
This type of engine provides immense power by expelling gas at high speed, which is essential to achieve supersonic speeds.

• Jet-powered cars are designed aerodynamically to reduce drag.
• They require advanced materials to handle the high temperatures and stresses encountered at high speeds.
• These cars are specifically developed for speed record attempts.

The use of a jet engine allows these cars to reach and potentially surpass the speed of sound, challenging the limits of current technology.

Ideal gas law

The ideal gas law is a fundamental principle in physics and chemistry that describes the behavior of ideal gases.
It is expressed as \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.
This law helps us understand the properties of gases under various conditions.

• It shows that as temperature increases, either pressure or volume must also increase.
• Used to estimate changes in gas behavior and the speed of sound in air.

Under conditions where air behaves like an ideal gas, we can predict how temperature variations affect sound speed, vital for calculating speeds required to break the sound barrier.

Breaking the sound barrier

Breaking the sound barrier means traveling faster than the speed of sound, known as Mach 1.
When an object, such as a jet car, breaks this barrier, it moves through the air at supersonic speeds.

• This can result in a sonic boom, a loud explosion-like sound.
• Requires overcoming aerodynamic drag, needing immense power output.
• The design must minimize air resistance for efficient high-speed travel.

Breaking the sound barrier is a significant achievement in speed records, demanding precise engineering and consideration of environmental factors like temperature.