Two trains approach each other at a railroad crossing. Train 1 travels 80 km/h and Train 2 travels 50 km/h. They both sound their horns before passing each other, emitting sound waves at 300 Hz. The speed of sound is 343 m/s. At what frequencies does a person waiting at the train tracks hear both train horns as they approach?
To find the frequencies that a person waiting at the train tracks hears both train horns as they approach, we need to consider the Doppler effect. The Doppler effect is the change in frequency or wavelength of a wave as an observer moves relative to the source of the wave.
Let's break down the problem step by step:
1. Determine the speed of each train relative to the person waiting at the train tracks:
- Train 1 travels at 80 km/h.
- Train 2 travels at 50 km/h.
2. Convert the speeds from km/h to m/s to match the units of the speed of sound:
- Train 1: 80 km/h = (80 * 1000) m / (60 * 60) s = 22.22 m/s
- Train 2: 50 km/h = (50 * 1000) m / (60 * 60) s = 13.89 m/s
3. Calculate the frequencies heard by the person as each train approaches using the Doppler effect formula:
- For the approaching train (Train 1, speed = 22.22 m/s):
- f1 = f * (v + v1) / (v + vs)
- v = speed of sound = 343 m/s
- f = emitted frequency = 300 Hz
- v1 = speed of Train 1 = 22.22 m/s
- vs = velocity of the observer = 0 (since the person waiting is stationary)
- For the approaching train (Train 2, speed = 13.89 m/s):
- f2 = f * (v + v2) / (v + vs)
- v = speed of sound = 343 m/s
- f = emitted frequency = 300 Hz
- v2 = speed of Train 2 = 13.89 m/s
- vs = velocity of the observer = 0 (since the person waiting is stationary)
4. Calculate the final frequencies heard by the person:
- For Train 1: f1 = 300 * (343 + 22.22) / (343 + 0) ≈ 308.52 Hz
- For Train 2: f2 = 300 * (343 + 13.89) / (343 + 0) ≈ 303.21 Hz
So, the person waiting at the train tracks will hear Train 1's horn around 308.52 Hz and Train 2's horn around 303.21 Hz as the trains approach each other.