Problem 57 - Physics

Question

A rocket, starting from rest, travels straight up with an acceleration of \(58.0 \mathrm{~m} / \mathrm{s}^{2}\). When the rocket is at a height of \(562 \mathrm{~m}\), it produces sound that eventually reaches a ground-based monitoring station directly below. The sound is emitted uniformly in all directions. The monitoring station measures a sound intensity \(I .\) Later, the station measures an intensity \(\frac{1}{3} I\). Assuming that the speed of sound is \(343 \mathrm{~m} / \mathrm{s},\) find the time that has elapsed between the two measurements.

Step-by-Step Solution

Answer

6.61 seconds have elapsed between the two measurements.

Step 1: Calculate Time to Reach Height

To calculate the time taken to reach a height of \( h = 562 \mathrm{~m} \), use the kinematic equation for uniformly accelerated motion: \[ h = \frac{1}{2} a t^2\]Plugging in the given values, we have \[ 562 = \frac{1}{2} \cdot 58 \cdot t^2\]Solving for \( t \), \[ t^2 = \frac{2 \times 562}{58} \Rightarrow t^2 = 19.379 \Rightarrow t = \sqrt{19.379} \approx 4.40\,\text{seconds}\]The rocket takes approximately \(4.40\) seconds to reach a height of \(562\, \mathrm{m}\).

Step 2: Calculate Time for Sound to Travel

The time it takes for the sound to travel from the rocket to the ground is calculated by using the speed of sound. Using the equation: \[ t_\text{sound} = \frac{h}{v_\text{sound}} = \frac{562\, \text{m}}{343\, \text{m/s}} \approx 1.64\,\text{seconds}\]So, the time taken for the sound to travel to the monitoring station is \(1.64\) seconds.

Step 3: Calculating Total Time for Initial Sound Detection

The total time to detect the first sound measurement at the ground is the sum of the time taken for the rocket to reach \(562\, \text{m}\) and the time for sound to travel to the ground.\[ t_1 = t_\text{rocket} + t_\text{sound} = 4.40 + 1.64 = 6.04\,\text{seconds}\]Thus, the initial sound is detected at approximately \(6.04\) seconds after the rocket starts.

Step 4: Calculate Distance for Lower Intensity

Sound intensity decreases with the square of the distance, according to the inverse square law:\[ I \propto \frac{1}{d^2} \]Given that intensity at the second time is \( \frac{1}{3} I \), the equation becomes:\[ \frac{1}{d_2^2} = \frac{1}{3} \cdot \frac{1}{d_1^2} \]Let \(d_1 = 562\) meters, then:\[ 3d_1^2 = d_2^2 \Rightarrow d_2 = \sqrt{3} \times 562 \approx 973.34 \,\text{meters}\]The sound is detected at this intensity from a distance of \(973.34\, \text{meters}\).

Step 5: Calculate Rocket Travel Time to New Height

The additional height the rocket has traveled can be calculated using:\[ 973.34 - 562 = 411.34 \,\text{meters}\]Using \( h = \frac{1}{2} a t^2 \) to find the additional time \[ 411.34 = \frac{1}{2} \times 58 \times t^2 \Rightarrow t^2 = \frac{411.34 \times 2}{58} = 14.18 \Rightarrow t = \sqrt{14.18} \approx 3.77\,\text{seconds}\]So, it takes approximately \(3.77\) seconds for this additional rocket height.

Step 6: Calculate Total Time for Second Detection

The total time taken till the sound of lower intensity reaches the ground is the sum of the previous travel time of \(6.04\) seconds, the new travel time, and the new sound travel time:\[ t_2 = 3.77\,\text{seconds (for rocket height)} + \frac{973.34}{343}\,\text{seconds (for sound travel)} \approx 2.84\]\[ t_2 = 6.04 + 3.77 + 2.84 = 12.65\,\text{seconds}\]Therefore, the second measurement is detected \(12.65\) seconds after the start.

Step 7: Calculate Time Elapsed Between Measurements

The time elapsed between the first and the second intensity measurement is the difference between the two total times calculated:\[ \Delta t = 12.65 - 6.04 = 6.61\,\text{seconds}\]Therefore, \(6.61\) seconds have elapsed between the two measurements.

Key Concepts

Understanding Uniform Acceleration

Uniform acceleration occurs when an object's velocity increases or decreases at a constant rate. It is a fundamental concept in kinematics, and it simplifies the calculation of motion scenarios, like a rocket launching upward. The formula used in such situations is derived from uniformly accelerated motion:

• \( h = \frac{1}{2} a t^2 \)

Here, \( h \) is the height, \( a \) is the acceleration, and \( t \) is the time taken. The equation shows the relationship between these variables when acceleration doesn't change over time.
For example, if a rocket starts from rest and moves upward with an acceleration of 58 m/s², this equation can help find out how long it takes to reach a certain height. Just plug in the known values for height and acceleration, solve for time \( t \), and you'll have how long the rocket took to ascend.

Exploring Sound Intensity

Sound intensity refers to the power per unit area carried by a sound wave. It determines how loud a sound is perceived to be and is measured in watts per square meter (W/m²). The intensity of sound at any point depends on the source of the sound and the distance from that source.

• The further away you are, the less intense the sound appears.

This is why loud sounds from distant sources may be hardly heard, while nearby quiet sounds are heard clearly. For example, when a rocket emits sound as it ascends, the intensity at any given point will change as the rocket moves further away.

Applying the Inverse Square Law

The inverse square law explains how a physical quantity like sound intensity decreases with increased distance from the source. Put simply, it states that the intensity of sound is inversely proportional to the square of the distance away from the sound source:

• \( I \propto \frac{1}{d^2} \)

This means that if you double the distance from the sound source, the intensity becomes one-fourth as strong. In our scenario, when the rocket sound's intensity is reduced to \( \frac{1}{3} \), the distance must increase by a factor of \( \sqrt{3} \).
This law helps in calculating how far the source has moved if the sound's intensity drops by a known ratio, such as when monitoring stations need to detect sound intensities.

Calculating Speed of Sound

Sound travels through the air at a known speed, commonly set at approximately 343 meters per second (m/s) at room temperature. This value of speed of sound is crucial when calculating how long it takes for sound to travel a specific distance.

• To find the time taken for sound to travel, use the formula: \( t_\text{sound} = \frac{h}{v_\text{sound}} \)

Here, \( h \) is the distance sound travels, and \( v_\text{sound} \) is the speed of sound.
In the case of the rocket, once the sound is generated, it travels downward and takes time to reach the ground station. By dividing the distance by the speed, you get the time it takes for the sound to be detected. Understanding this principle helps in synchronizing events over distances, such as the delay between seeing a rocket's launch and hearing it launch.