On a highway, two buses A and B are running at the same velocity of magnitude $$30$$ $$ms-1$$. The brakes caused a deceleration of $$307ms$$−$$2$$ in bus A and that of bus B is $$3$$ $$ms-2$$. In an emergency when driver of the front car applies brakes, immediately its rear light turns red and braking begins. In response, driver of the rear bus also applies brakes to avoid a collision with the front bus. Every driver takes $$1$$ s to apply the brakes after he saw a need for it. If bus A ahead of bus B, then the minimum separation between the buses before driver of bus A applies the brake is x. If bus B is running ahead of bus A, then the minimum separation between the buses before the driver of bus B applies brake is $$x2$$. The value of $$x13x2is$$ $$_______$$

Question

On a highway, two buses A and B are running at the same velocity of magnitude $$30$$ $$ms-1$$. The brakes caused a deceleration of $$307ms$$−$$2$$ in bus A and that of bus B is $$3$$ $$ms-2$$. In an emergency when driver of the front car applies brakes, immediately its rear light turns red and braking begins. In response, driver of the rear bus also applies brakes to avoid a collision with the front bus. Every driver takes $$1$$ s to apply the brakes after he saw a need for it. If bus A ahead of bus B, then the minimum separation between the buses before driver of bus A  applies the brake is  x. If bus B is running ahead of bus A, then the minimum separation between the buses before the driver of bus B applies brake is $$x2$$. The value of $$x13x2is$$ $$_______$$

Infinity Learn · Accepted Answer

For $$x1$$, If collision does not like place. The final velocities of both should be same.∴ $$vf=30$$−$$307t=30$$−$$3(t$$−$$1)$$∴ $$307t=3t$$−$$3$$ or $$307t$$−$$3t=$$−$$3$$ Or  $$9t7=$$−$$3$$⇒$$t=$$−$$219The$$ negative of t indicates that both come to rest before collision. The distance moved by bus B before coming to rest $$iss1=30$$×$$1+3022$$×$$3=30+150=180mThe$$ distance moved by bus A before coming to rest is $$s2=3022$$×$$307=30$$×$$72=105m$$∴ $$x1=s1$$−$$s2=180$$−$$105=75m$$ For  $$x2$$,$$vf=30$$−$$307(t$$−$$1)=30$$−$$3t$$ Or  $$307t$$−$$307=3t$$ or $$97t=307$$∴ $$t=103sThe$$ positive of t indicates that before collision, velocities of both buses are $$non-zero$$ but same.So, the distance moved by A before collision $$iss1=30$$×$$1+30$$×$$103$$−$$1$$−$$12$$×$$307103$$−$$12s1=2653ms2=30$$×$$103$$−$$12$$×$$3$$×$$1032=100$$−$$503=2503x2=s1$$−$$s2=2653$$−$$2503=153=5m$$ Or $$x13x2=753$$×$$5=5$$

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