have a couple of questions about my homework and it's due at midnight tonight:
1. On a hot summer day, the temperature of air in Arizona reaches 115°F. What is the speed of sound in air at this temperature? (Enter your answer to four significant figures. The speed of sound at 0°C is 331 m/s. Use the conversion 0°C = 273 K as necessary.)
I changed F to K (got 315.928) and I plugged it in the appropriate equation:
331m/s(315.928/273)^(1/2) and I got 356.1. This is not right what do I do?
2.A dolphin located in sea water at a temperature of 25°C emits a sound directed toward the bottom of the ocean 105 m below. How much time passes before it hears an echo?
I did two different equations: dividing 105 by the temp in K, then I tried 331(298K/273 k)^(1/2) then I multiplied it by to (to denote that the sound had to return back) and got 0.607 but that wasn't right.
For Question 1: Calculating the speed of sound at a given temperature requires using the equation:
v = v0 * sqrt(T)
Where:
- v is the speed of sound at the given temperature
- v0 is the speed of sound at 0°C (which is given as 331 m/s)
- T is the temperature in Kelvin
To convert Fahrenheit (F) to Kelvin (K), you correctly used the formula: K = (F - 32) * 5/9 + 273.15. However, you made a mistake in your calculation. Let's re-calculate it step by step:
1. Convert 115°F to K:
K = (115 - 32) * 5/9 + 273.15
K = 46.11 + 273.15
K ≈ 319.26
2. Substitute the value of T into the equation:
v = 331 * sqrt(319.26/273)
v ≈ 331 * sqrt(1.171)
v ≈ 331 * 1.080
v ≈ 357.48
Rounded to four significant figures, the speed of sound in air at 115°F is approximately 357.5 m/s. So, the correct answer should be 357.5.
Now, let's move on to Question 2:
To determine the time it takes for the dolphin to hear an echo, we need to consider the speed of sound and the distance the sound travels. Let's break it down:
1. Convert the temperature in Celsius (°C) to Kelvin (K):
The temperature is given as 25°C, so K = 25 + 273 = 298 K.
2. Calculate the time it takes for the sound to travel 105 m (one way):
Using the equation:
t = d / v
Where:
- t is the time (in seconds)
- d is the distance (in meters)
- v is the speed of sound (in m/s)
Substituting the known values:
t = 105 / (331 * sqrt(298/273))
t ≈ 105 / (331 * 1.015)
t ≈ 105 / 336.165
t ≈ 0.3126
To account for the return trip, we need to double the time:
Total time = 2 * 0.3126 ≈ 0.6252
Rounded to the appropriate significant figures, the time it takes for the dolphin to hear an echo is approximately 0.625 seconds.
Please note that rounding may vary based on your instructor's requirements, so make sure to follow any specific guidelines provided.