Problem 61 - Algebra and Trigonometry

Question

An explosion is recorded by two microphones that are 1 mile apart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2}\). Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.

Step-by-Step Solution

Answer

The explosion occurred at a point on the line of \(M_{1}\) and \(M_{2}\) which is at a distance between 2200 ft and 7480 ft from \(M_{1}\). Therefore the possible explosion points in relation to \(M_{1}\) and \(M_{2}\) on a coordinate plane would be any point (x,0) where \(2200ft <= x <= 7480ft \).

Step 1: Understand the problem

The sound from the blast travels at 1100 feet per second and the two microphones are 1 mile (or 5280 feet) apart. From the problem, it's known that microphone \(M_{1}\) received the sound 2 seconds earlier than microphone \(M_{2}\). The task at hand is to calculate the possible locations of the explosion.

Step 2: Apply the velocity and time to calculate distances from the explosion

Now, in the time microphone \(M_{2}\) hasn't received the sound yet, the wave from the explosion has travelled a further \(2s * 1100ft/s = 2200ft\). Therefore, when \(M_{2}\) starts recording, the sound is at any point in a circle with radius 2200 ft centered on \(M_{1}\). When \(M_{2}\) finishes recording, the sound has travelled \(5280ft\) further, so then it is at any point in a circle with radius \(7480ft (5280 + 2200)\) centered on \(M_{1}\). The possible locations of the explosion are points which are at the same time on the line of \(M_{1}\) and \(M_{2}\) (which are 5280 ft apart) and in the ring between the two circles centered on \(M_{1}\).

Step 3: Calculate and interpret the possible locations

The explosion therefore, was somewhere on the line of \(M_{1}\) and \(M_{2}\) at a distance of between 2200 ft from \(M_{1}\) and 7480 ft from \(M_{1}\). If \(M_{1}\) and \(M_{2}\) are at points (0,0) and (5280,0) respectively, the explosion was at a point (x,0), where \(2200ft <= x <= 7480ft \).

Key Concepts

Explosion Detection

To understand how explosions are detected using sound, you need to imagine how sound travels. An explosion creates a wave of sound that moves through the air. This wave gradually reaches different locations, including microphones strategically placed at different points.

When the wave from an explosion hits a microphone, its sensor records the sound. If there are multiple microphones, like in our exercise with microphones at different locations, each detects the sound at slightly different times. The time difference helps us deduce where the explosion occurred because sound travels at a constant speed.

Thus, detecting time differences in when the sound reaches each microphone is key to finding the explosion's location.

Microphone Distance

When you're thinking about microphone distance, think about how far apart the devices are placed from each other. In our exercise, the microphones are distanced 1 mile apart, which equals 5280 feet.

This measurement is crucial because it determines the additional distance the sound wave travels to reach the second microphone from the first one. Knowing how far the microphones are from each other allows us to use this distance in calculations that reveal the potential location of the explosion.

The result is a clearer understanding of how varying microphone placements affect the ability to detect sound accurately.

Speed of Sound

The speed of sound plays a vital role in calculating the time it takes for sound to travel between locations. For our solution, we assume sound travels at 1100 feet per second. This is important because:

• It is the rate at which the sound created by the explosion travels through the air.
• A known speed allows us to compute how far sound can travel in a set period.

Multiply the speed of sound by the time difference to find out how far the sound wave has moved. Comprehending the speed of sound lays the groundwork for calculations determining where the sound originated from.

In this context, since the sound wave reaches the first microphone 2 seconds before the second, we multiply 2 seconds by the speed of sound (1100 feet/second) to understand the spatial relationship of the microphones to the sound source.

Time Delay

Time delay is the difference in time it takes for the sound to reach each microphone. In the explosion scenario, microphone 1 (M1) receives the sound 2 seconds ahead of microphone 2 (M2).

This delay is critical in pinpointing where the explosion happened. Using the speed of sound, we calculate that this 2-second delay indicates an additional distance of 2200 feet (2 seconds * 1100 feet/second) between the microphones and the sound source.

Understanding and accurately measuring this time delay helps map out a more precise location of the explosion, shedding light on the corresponding distance and ensuring all results align with the laws of physics.