Two trains on separate tracks move toward one another. Train 1 has a speed of 125 km/h and train 2 a speed of 100.0 km/h. Train 2 blows its horn, emitting a frequency of 500 Hz. What is the frequency heard by the engineer on train 1?
(Assume the speed of sound is 345 m/s.)
To find the frequency heard by the engineer on train 1, we can use the concept of the Doppler effect. The Doppler effect describes the change in frequency of a wave (in this case, sound waves) due to relative motion between the source of the wave (train 2) and the observer (train 1).
The formula for the Doppler effect is:
f' = (v + vo)/(v + vs) * f
Where:
f' is the frequency heard by the observer (train 1),
v is the speed of sound,
vo is the velocity of the observer (train 1),
vs is the velocity of the source (train 2), and
f is the frequency of the source (train 2).
In this case:
v = 345 m/s (speed of sound),
vo = 125 km/h = 125 * (1000 m/1 km) / (3600 s/1 h) = 34.722 m/s (velocity of train 1),
vs = -100.0 km/h = -100.0 * (1000 m/1 km) / (3600 s/1 h) = -27.778 m/s (velocity of train 2),
f = 500 Hz (frequency of train 2).
Using the provided values, we can substitute them into the Doppler effect formula:
f' = (v + vo)/(v + vs) * f
f' = (345 + 34.722)/(345 - 27.778) * 500
f' = 379.722/317.222 * 500
f' = 1.196 * 500
f' = 598 Hz
Therefore, the frequency heard by the engineer on train 1 is approximately 598 Hz.