a train whistle is heard at 305 hz as the train approaches town. the train cuts its speed in half as it nears the station, and the sound of the whistle is then 288 hz. what is the speed of the train after slowing down? speed of sound is 343 m/s.

The rate at which the location of an object shifts in any direction. Speed is defined as the ratio of the distance traveled to the time required to cover that distance.

Briefing:

[tex]f=f_0 \frac{v+v_o}{v-v_a}[/tex]

f₀: is the emitted frequency

f: is the frequency the observer

v: speed of sound

v₀: is the speed of the observer = 0 (it is heard in the town)

v[tex]_s[/tex]: is the speed of the source =?

Before slowing down, the train's frequency is determined by:

[tex]f_b=f_0 \frac{v}{v-v_{\Delta_b}}[/tex](1)

Frequency of the train after slowing :

[tex]f_a=f_0 \frac{v}{v-v_{s_a}}[/tex](2)

Dividing equation (1) by (2) we have:

[tex]\frac{f_b}{f_a}=\frac{f_0 \frac{v}{v-v_{d_b}}}{f_0 \frac{v}{v-v_{s_a}}}[/tex]

[tex]\frac{f_b}{f_a}=\frac{v-v_{v_a}}{v-v_{s_b}}[/tex](3)

Additionally, we are aware that the train slows to half its starting speed, so:

[tex]v_{s_b}=2 v_{s_a}[/tex]

Entering equation (4) into (3) we have:

[tex]\frac{305 \mathrm{~Hz}}{288 \mathrm{~Hz}}=\frac{343 \mathrm{~m} / \mathrm{s}-v_{s_a}}{343 \mathrm{~m} / \mathrm{s}-2 v_{s_a}}[/tex]

After the train has slowed down, we can determine its speed:

[tex]v_{s_a}[/tex] = 11.64 m/s

The train's speed just before it begins to slow down is:

[tex]v_{s_b}[/tex] = 11.64 m/s * 2 = 23.28 m/s

The speed of the train before and after slowing down is 23.28 m/s and 11.64 m/s, respectively.

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