1.1. A train is travelling at 45 m s⁻¹ and blows a whistle at a frequency of 750 Hz. A person is waiting at a level crossing for the train to pass. Take the speed of sound in air as 340 m s⁻¹.

Calculate the frequency observed by the person: 

(i) In still air.

[1]

(ii) When the wind is blowing at 10 m s⁻¹ towards the train and away from the person.

[2]

1.2. After the train passes the level crossing, it slows down as it approaches a tunnel. Wind is funnelled down the tunnel towards the train reaching a velocity of 27 m s⁻¹. The observer at the level crossing hears the whistle at a frequency of 729 Hz. 

Calculate the new speed of the train.

1.3. After exiting the tunnel, the train stops at a station. The observer from part (a) is now walking to the platform to catch the train at a speed of 1.5 m s⁻¹. The human ear can distinguish between frequency differences of around 3.6 Hz.

The Doppler equation for a moving observer is:

${f}_{o}={f}_{s}(\frac{v\pm{v}_{o}}{v})$

Show that the change in frequency of the train's whistle, as heard by the walking observer, can be considered negligible.

1.4. Once the train has left the station, it travels down a line with two tracks. Another train is approaching on the opposite track with a speed of 7.6 m s⁻¹. The speed of the departing train is 83% of the speed of the second train.

When both the observer and source are moving, the following equation can be used to determine the new frequency:

${f}_{o}={f}_{s}(\frac{v+{v}_{o}}{v-{v}_{s}})$

Calculate the frequency of this same whistle as observed by a passenger on the approaching train as a percentage of ${f}_{s}$.