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A train blowing its whistle moves with a constant velocity u away from an observer on the ground. The ratio of the actual frequency of the whistle to that measured by the observer is found to be 1.2. If the train is at rest and the observer moves away from it at the same velocity, then the ratio would be
  • a)
    0.51
  • b)
    1.25
  • c)
    1.52
  • d)
    2.05
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A train blowing its whistle moves with a constant velocity u away from...
If the train is going away from the observer, the apparent frequency is
In the second case, the apparent frequency is
Now, from (i) we have
Using this in (ii), we get
Hence the correct choice is (2).
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Community Answer
A train blowing its whistle moves with a constant velocity u away from...
Given:
Velocity of train, u
Ratio of actual frequency to observed frequency, k = 1.2

To find:
Ratio of actual frequency to observed frequency when the observer moves away from the train at the same velocity.

Solution:
Let the frequency of the whistle emitted by the train be f.
Let the frequency of the whistle observed by the observer be f'.
Let the velocity of sound in air be v.

When the train is at rest and the observer is stationary, the observed frequency is given by,

f' = f

When the train is moving away from the observer, the observed frequency is given by the Doppler's effect formula,

f' = f × (v + u) / (v + us)

where s is the distance between the train and the observer.

Given that the ratio of actual frequency to observed frequency is 1.2, we have

f / f' = 1.2

Substituting the expressions for f and f', we get

f / (f × (v + u) / (v + us)) = 1.2

Simplifying, we get

v + us = 1.2(v + u)

us = 0.2v / 0.8

us = 0.25v

Now, when the observer moves away from the train with the same velocity u, the observed frequency is given by

f'' = f × (v + u) / (v + u + u)

f'' = f × (v + u) / (v + 2u)

The ratio of actual frequency to observed frequency is given by

f / f'' = f / (f × (v + u) / (v + 2u))

f / f'' = (v + 2u) / (v + u)

Substituting the value of u, we get

f / f'' = (v + 2u) / (v + u)
f / f'' = (v + 2(0.25v)) / (v + 0.25v)
f / f'' = 5 / 4
f'' / f = 4 / 5

Hence, the ratio of actual frequency to observed frequency when the observer moves away from the train at the same velocity is 4/5 or 0.8, which is approximately equal to 1.25 (option B).
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A train blowing its whistle moves with a constant velocity u away from an observer on the ground. The ratio of the actual frequency of the whistle to that measured by the observer is found to be 1.2. If the train is at rest and the observer moves away from it at the same velocity, then the ratio would bea)0.51b)1.25c)1.52d)2.05Correct answer is option 'B'. Can you explain this answer?
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