A bus B is moving with a velocity vB in the positive $$x-direction$$ along a road as shown in the figure. A shooter ‘S ’ is at a distance l from the road. He has a detector which can detect signals only of frequency $$1500Hz$$. The bus blows horn of frequency $$1000$$ Hz. When the detector detects a signal the shooter immediately shoots towards the road along SC and the bullet hits the bus. Find the velocity of the bullet if velocity of sound in air is v $$=$$ $$340$$ $$m/s$$ and $$VBV=233$$

Question

A bus B is moving with a velocity vB in the positive $$x-direction$$ along a road as shown in the figure. A shooter ‘S ’ is at a distance l from the road. He has a detector which can detect signals only of frequency $$1500Hz$$. The bus blows horn of frequency $$1000$$ Hz. When the detector detects a signal the shooter immediately shoots towards the road along SC and the bullet hits the bus. Find the velocity of the bullet if velocity of sound in air is v $$=$$ $$340$$ $$m/s$$ and $$VBV=233$$

Infinity Learn · Accepted Answer

Let the bus be at O when it sends a signal that is detected by the detector as of frequency $$=$$ $$1500$$ Hz∴ f′$$=vv$$−vBcos⁡θ×$$1000=1500$$∴ $$cos$$⁡θ$$=32$$⇒θ$$=30$$∘By the time signal reaches at S the bus reaches at D.Let this time be $$t0$$∴ $$t0=OSv=$$∣cosec⁡θv…………….(1)Now$$ man fires and the bullet reaches C in time $$t1$$ $$(say). In the same time bus moves from D to C∴ $$t1=1u$$; where $$u=$$ speedofbullet Also, OD $$+$$ DC $$=$$ lcotθ    $$vBt0+vBt1=Icot$$⁡θ    vBIcosec⁡θ$$v+vB(l/u)=Icot$$⁡θ    $$233$$×$$2+233vu=3$$∴     $$vu=52$$∴     $$u=25v=25$$×$$340=00136.00m/s$$

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