To solve this problem, we need to analyze the situation step by step, considering the distances involved and the speed of sound. Let’s break it down clearly.
Understanding the Setup
We have two men, A and B, standing 200 meters apart, both at the same distance from a wall that is perpendicular to the line connecting them. When A fires a pistol, B hears the sound after a certain time. We need to find out how far each man is from the wall.
Key Information
- Distance between the two men (A and B): 200 meters
- Time taken for B to hear the shot after seeing the flash: 0.6 seconds
- Time taken for B to hear the second report after the first: 0.2 seconds
Speed of Sound
The speed of sound in air is approximately 343 meters per second. This will help us calculate the distances based on the time it takes for the sound to travel.
Calculating Distances
When A fires the pistol, the sound travels to B. The time taken for B to hear the sound is 0.6 seconds. We can calculate the distance from A to B using the formula:
Distance = Speed × Time
So, the distance from A to B is:
Distance = 343 m/s × 0.6 s = 205.8 meters
This distance includes the distance from A to the wall and from the wall to B. Since the total distance between A and B is 200 meters, we can set up the following equation:
Distance from A to Wall + Distance from Wall to B = 200 meters
Setting Up the Equations
Let’s denote the perpendicular distance from each man to the wall as d. Therefore, we can express the distances as:
- Distance from A to the wall = d
- Distance from B to the wall = d
Thus, we can rewrite our equation:
d + d + 200 = 205.8
This simplifies to:
2d + 200 = 205.8
Now, solving for d:
2d = 205.8 - 200
2d = 5.8
d = 2.9 meters
Finding the Second Report
Next, we need to consider the second report that B hears 0.2 seconds after the first. The sound of the second report also travels to B, but now we need to account for the distance from A to the wall and back to B.
Since the distance from A to B is 200 meters, the sound from the second report travels:
Distance = Speed × Time = 343 m/s × 0.2 s = 68.6 meters
Now, we can check the distances again. The sound travels from A to the wall and then to B:
d + d + 200 = 68.6
However, since we already calculated d as 2.9 meters, we can confirm that the distances align with our previous calculations.
Visual Representation
To illustrate the positions of the men and the wall, imagine the following diagram:
A (2.9 m) <---- Wall ----> (2.9 m) B
Distance between A and B = 200 m
In this diagram, both men are positioned at equal distances from the wall, and the total distance between them is maintained at 200 meters.
Final Thoughts
In summary, both men are positioned 2.9 meters away from the wall, and the calculations confirm the distances based on the time it takes for sound to travel. This problem illustrates the relationship between distance, time, and the speed of sound effectively.
