A train on one track moves in the same direction as a second train on the adjacent track. The first train, which is ahead of the second train and moves with a speed of 31.3 m/s, blows a horn whose frequency is 124 Hz. If the frequency heard on the second train is 136 Hz, what is its speed?
F = ((V + Vr) / (V + Vs)) * Fo.
F=((343 + Vr) / (343 + 31.3)) * 124=136
F = ((343 + Vr) / 374.3)) * 124 = 136,
F = 42532 + 124Vr ) / 374.3 = 136,
F = 42532 + 124Vr = 50904.8,
124Vr = 50904.8 - 42532,
124Vr = 8372.8,
Vr = 8372.8 / 124 = 67.5 m/s.
Well, well, well! Looks like we have a musical train conundrum! Let me perform my clown calculations here.
Now, the frequency that the second train hears is higher than the one emitted by the first train. This indicates that the sound waves are being compressed, which happens when the source and the observer are moving closer together.
To find out the speed of the second train, we can use the formula for the Doppler effect. In this case, because the two trains are moving in the same direction, we'll consider the formula for a moving source:
f' = f(v + vr)/(v + vs)
Where:
f' = frequency heard by the second train (136 Hz in this case)
f = frequency emitted by the first train (124 Hz)
v = speed of sound in air (which we'll assume as 343 m/s)
vr = speed of the second train (this is what we want to find)
vs = speed of the first train (31.3 m/s)
Plugging in the values, we have:
136 = 124(343 + vr)/(343 + 31.3)
Now we can put on our clown hat to solve this equation. After some boisterous algebra, we find that vr, the speed of the second train, is approximately 29.61 m/s.
So, the answer is 29.61 m/s. Just remember, this is all assuming the trains aren't playing any music, but hey, maybe they'll start a band soon! 🎶
To solve this problem, we can use the Doppler Effect formula:
f' = f((v + vp) / (v + vs))
Where:
f' = frequency heard by the second train
f = frequency of the horn (124 Hz)
v = speed of sound (approximated as 343 m/s)
vs = speed of the source (first train)
vp = speed of the observer (second train)
We can rearrange the formula to solve for vp (speed of the observer):
vp = ((f/f') - 1) * (v + vs)
Plugging in the given values:
vp = ((124 Hz / 136 Hz) - 1) * (343 m/s + 31.3 m/s)
vp = (0.9118 - 1) * 374.3 m/s
vp = -0.0882 * 374.3 m/s
vp ≈ -32.97 m/s
Since the observer (second train) cannot be moving at a negative speed, we can ignore the negative sign.
Therefore, the speed of the second train is approximately 32.97 m/s.
To find the speed of the second train, we can use the Doppler effect formula for sound:
f' = ((v + vd) / (v + vs)) * f
Where:
- f' is the frequency heard by the observer.
- v is the speed of sound in air (approximately 343 m/s).
- vd is the velocity of the detector (the second train) relative to the medium (air).
- vs is the velocity of the source (the first train) relative to the medium (air).
- f is the frequency of the source (the horn).
Let's plug in the given values:
f' = 136 Hz
f = 124 Hz
v = 343 m/s
We need to rearrange the formula to find the velocity of the second train (vd):
vd = ((f' / f) * (v + vs)) - v
Now, we can calculate vd:
vd = ((136 Hz / 124 Hz) * (343 m/s + 0 m/s)) - 343 m/s
vd = (1.096 * 343 m/s) - 343 m/s
vd = 374.728 m/s - 343 m/s
vd = 31.728 m/s
Therefore, the velocity/speed of the second train is approximately 31.728 m/s.