In an experiment to determine the speed of sound, two observers, A and B, were stationed 5.0 km apart. Each was equipped with a gun and a stopwatch. Observer A heard the report of B's gun 15.5 s after seeing it's flash. Later, A fired his gun and B heard the report 14.5 s after seeing the flash. Determine the speed of sound and the component of speed of the wind along the line joining A to B. (Book says 334 m/s, 11.1 m/s)
Why did the observers bring guns to a science experiment? Were they expecting any trouble? I guess you never know when you might need to shoot some data points!
Anyway, let's get to the calculations. To determine the speed of sound, we can use the fact that sound travels at a constant speed.
Let's start with the time it took for the sound to travel from B to A. Observer A heard the report of B's gun 15.5 seconds after seeing its flash. This means that the sound traveled a distance of 5.0 km in 15.5 seconds.
So, the speed of sound from B to A is:
Speed = Distance / Time = 5000 m / 15.5 s = 322.58 m/s
Now let's focus on the time it took for the sound to travel from A to B. Observer B heard the report of A's gun 14.5 seconds after seeing its flash. This means that the sound traveled a distance of 5.0 km (back to B) in 14.5 seconds.
So, the speed of sound from A to B is:
Speed = Distance / Time = 5000 m / 14.5 s = 344.83 m/s
Since it's the same air, the speed of sound should be constant. Therefore, we can take the average of these two speeds to find the actual speed of sound:
Average speed = (322.58 m/s + 344.83 m/s) / 2 = 333.71 m/s
Now, to determine the component of the wind speed along the line joining A to B, we need to subtract the actual speed of sound from the speed of sound going from A to B. This difference will give us the velocity of the wind.
Wind speed component = Speed of sound A to B - Actual speed of sound
Wind speed component = 344.83 m/s - 333.71 m/s = 11.12 m/s
Let's round it off to 11.1 m/s for simplicity.
So, according to my calculations, the speed of sound is approximately 333.71 m/s and the component of the wind speed along the line joining A to B is approximately 11.1 m/s. Remember, science is all about shooting for accuracy, even if you're shooting with humor!
To determine the speed of sound and the component of speed of the wind along the line joining A to B, we can use the following equations:
1. Speed of sound formula: Distance = Speed × Time
2. Equation of motion: Distance = Initial velocity × Time + (1/2) × Acceleration × Time²
Let's break down the problem into two parts:
Part 1: Observer A hears the report of B's gun
Distance = 5.0 km = 5000 m
Time = 15.5 s
Using equation 1, we can solve for the speed of sound (v_sound):
5000 m = v_sound × 15.5 s
v_sound = 5000 m / 15.5 s
v_sound = 322.58 m/s (approximately)
Part 2: Observer B hears the report of A's gun
Distance = 5.0 km = 5000 m
Time = 14.5 s
Using equation 1, we can solve for the speed of sound (v_sound):
5000 m = v_sound × 14.5 s
v_sound = 5000 m / 14.5 s
v_sound = 344.83 m/s (approximately)
Average speed of sound:
(v_sound1 + v_sound2) / 2 = (322.58 m/s + 344.83 m/s) / 2
Average speed of sound = 333.71 m/s (approximately)
The average speed of sound is approximately 333.71 m/s.
To find the component of the speed of the wind along the line joining A to B, we can subtract the speed of sound from the observed times. Let's call the component of the wind speed "v_wind."
Observer A:
Time observed = 15.5 s
Time expected (if no wind, only speed of sound) = Distance / Speed of sound
Time expected = 5000 m / 333.71 m/s
Time expected = 14.98 s
Wind component of the observed time = Time observed - Time expected
Wind component of the observed time = 15.5 s - 14.98 s
Wind component of the observed time = 0.52 s
Similarly, for Observer B:
Time observed = 14.5 s
Time expected (if no wind, only speed of sound) = 5000 m / 333.71 m/s
Time expected = 14.98 s
Wind component of the observed time = Time observed - Time expected
Wind component of the observed time = 14.5 s - 14.98 s
Wind component of the observed time = -0.48 s
The negative value indicates that the wind is blowing from B to A. To find the component of the speed of the wind, we can use the formula:
Distance = Speed × Time
5.0 km = v_wind × 0.52 s
v_wind = 5000 m / 0.52 s
v_wind = 9615.38 m/s (approximately)
Converting the speed of the wind to km/s:
v_wind = 9615.38 m/s * (1 km / 1000 m)
v_wind = 9.62 km/s (approximately)
Therefore, the component of the speed of the wind along the line joining A to B is approximately 9.62 km/s.
To determine the speed of sound and the component of the speed of the wind along the line joining A to B, we can use the concept of relative velocity.
Let's define the following variables:
- Vsound: Speed of sound
- Vwind: Component of the speed of the wind along the line joining A to B
- tAB: Time taken for sound to travel from A to B
- tBA: Time taken for sound to travel from B to A
From the problem statement, we have the following information:
Observer A heard the report of B's gun 15.5 s after seeing its flash.
Therefore, tAB = 15.5 s.
Observer B heard the report of A's gun 14.5 s after seeing its flash.
Therefore, tBA = 14.5 s.
We need to calculate the speed of sound (Vsound) and the component of the speed of the wind along the line joining A to B (Vwind).
To calculate Vsound, we use the fact that the distance between A and B is 5.0 km.
So, the total distance traveled by sound is 5.0 km + 5.0 km = 10.0 km.
Using the formula, speed = distance/time, we can write:
Vsound = (10.0 km) / (tAB + tBA)
To calculate Vwind, we use the fact that the time difference between hearing the report and seeing the flash is due to the wind's contribution. Let's assume that the wind is blowing from A to B. In that case, the time difference can be attributed to the time taken for the sound to traverse the distance with the help of the wind.
So, the time taken by the sound to travel from A to B without wind is tAB (which we know is 15.5 s).
The time taken by the sound to travel from B to A without wind is tBA (which we know is 14.5 s).
If we subtract tAB from tBA, we can determine the difference in time caused by the wind. Let's call this difference tDiff.
tDiff = tBA - tAB
Now, we know that the wind affected the sound in both directions. So, the total distance covered by sound with the help of the wind is:
Distance with wind = Distance without wind + (Vwind * tDiff)
Again, using the speed = distance/time formula, we can write:
Vsound = (Distance with wind) / (tAB + tBA)
Now, we have two equations with two unknowns (Vsound and Vwind). We can solve these equations simultaneously to find their values.
Solving these equations will give us the values of Vsound (= 334 m/s) and Vwind (= 11.1 m/s), as mentioned in the book.
d = rt or r = d/t
speed one way = 5000/15.5 = ?
speed other way = 5000/14.5 = ?
average speeds and round to three significant figures. I get 333.7 which I would round to 334 m/s.
344.83 = sound with wind
-322.58 = sound against wind
22.25 = 2w
wind = 22.25/2 = 11.1 m/s