Question

A rocket is moving at a speed of \(130 \mathrm{~m} / \mathrm{s}\) towards a stationary target. While moving, it emits a wave of frequency \(800 \mathrm{~Hz}\). Calculate the frequency of the sound as detected by the target. (Speed of wave \(=330 \mathrm{~m} / \mathrm{s}\) ) (A) \(1320 \mathrm{~Hz}\) (B) \(2540 \mathrm{~Hz}\) (C) \(1270 \mathrm{~Hz}\) (D) \(660 \mathrm{~Hz}\)

Short answer

The frequency of the sound as detected by the target is (A) \(1320 \mathrm{~Hz}\).

Step-by-step solution

Understand the Doppler effect formula for sound waves

The Doppler effect formula for sound waves is given by: \[f' = f \frac{v + v_0}{v + v_s}\] Here, \(f'\) = Apparent frequency detected by the target \(f\) = Source frequency (800 Hz) \(v\) = Speed of the sound wave (330 m/s) \(v_0\) = Speed of the target (0 m/s, as it is stationary) \(v_s\) = Speed of the source (the rocket), (positive when the source moves away from the observer and negative when it moves towards)

Plug the given values into the formula

We know the source frequency, speed of the sound wave, and speed of the rocket, so we can plug these values into the formula. Since the rocket is moving towards the target, the speed of the source will be negative. Hence, \(v_s = -130 \mathrm{~m / s}\).

Calculate the apparent frequency

Now, we can plug all these values into the Doppler effect formula and find the apparent frequency detected by the target. \[f' = 800 Hz \frac{330 m/s + 0}{330 m/s - 130 m/s}\]

Simplify and find the answer

Simplifying the equation, we get \[f' = 800 Hz \frac{330 m/s}{200 m/s}\] \[f' = 800 Hz \cdot \frac{33}{20}\] \[f' = 1320 Hz\] So, the frequency of the sound as detected by the target will be \(1320 Hz\). Therefore, the correct answer is (A) \(1320 \mathrm{~Hz}\).

Key concepts

Rocket Motion

Rocket motion involves understanding how a rocket moves through space or air. Rockets are vehicles that operate on the principle of Newton's third law, which states that for every action, there's an equal and opposite reaction. In the context of the problem, the rocket is moving at a steady speed of 130 meters per second. It's important to note that this speed is quite significant, as it affects the frequency of sound waves emitted by the rocket. When a rocket moves towards a target, the sound waves become compressed, leading to an increase in frequency—a phenomenon known as the Doppler effect. Understanding this motion helps to predict how sound frequencies will be perceived by stationary observers as the rocket approaches them.

Frequency Calculation

Frequency calculation is crucial in problems involving the Doppler effect. The frequency of a wave refers to how often the wave cycles occur in one second. In this exercise, a sound wave emitted by the rocket has a source frequency of 800 Hz. The purpose of the calculation is to determine the new frequency, known as the apparent frequency, as detected by a stationary target. Using the Doppler effect formula: \[f' = f \frac{v + v_0}{v + v_s}\]We can substitute the known values:

• Source frequency \(f = 800\) Hz
• Speed of sound \(v = 330\) m/s
• Target speed \(v_0 = 0\) m/s (stationary)
• Rocket speed \(v_s = -130\) m/s (negative because it moves toward the target)

By inserting these into the formula and simplifying, you get the apparent frequency \(f' = 1320\) Hz.

Wave Speed

Wave speed is a fundamental component of wave mechanics and refers to how fast a wave propagates through a medium. In this scenario, the speed of sound is given as 330 meters per second. This is the speed at which sound waves travel through air.
Sound waves are longitudinal waves, which means that their oscillations happen in the same direction as their motion. The speed of sound can vary based on factors like temperature, pressure, and the medium through which it travels. However, for this problem, we're assuming standard conditions, keeping the wave speed constant. This wave speed is crucial in applying the Doppler effect formula to find the new frequency as perceived by a stationary observer.

Sound Waves

Sound waves are vibrations that travel through air or another medium. They are a type of mechanical wave that requires a material medium for propagation.
In this exercise, sound waves are generated by the motion of the rocket and carry energy through the air reaching the target. Sound waves have characteristics such as frequency, wavelength, and amplitude, which determine aspects like pitch and volume.
The Doppler effect primarily alters the frequency of the sound waves as observed by a listener, which in this case is a stationary target. As the rocket approaches the target, sound waves are compressed, increasing their frequency and resulting in a higher pitch. Understanding these properties of sound waves helps in comprehending how and why the observed frequency changes as different conditions and relative motion come into play.