A train with $$cross-sectional$$ area St is moving with speed 𝑣𝑡 inside a long tunnel of $$cross-sectional$$ area $$S0$$ (S0$$ $$=$$ $$4St). Assume that almost all the air (density$$ ρ$$) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $$p0$$. If the pressure in the region between the sides of the train and the tunnel walls is p, then $$p0$$−$$p=72N$$ρ$$vt2$$. The value of N $$is_________$$

Question

A train with $$cross-sectional$$ area St is moving with speed 𝑣𝑡 inside a long tunnel of $$cross-sectional$$ area $$S0$$ (S0$$ $$=$$ $$4St). Assume that almost all the air (density$$ ρ$$) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $$p0$$. If the pressure in the region between the sides of the train and the tunnel walls is p, then $$p0$$−$$p=72N$$ρ$$vt2$$. The value of N $$is_________$$

Infinity Learn · Accepted Answer

Applying Bernoulli's $$equationP0+12$$ρ$$vt2=P+12$$ρ$$v2P0$$−$$P=12$$ρ$$v2$$−$$vt2$$….. (i) From equation of continuity Also, $$4Stvt=v$$×$$3St$$⇒$$v=43vt$$ … (ii)  From (i) and (ii) $$P0$$−$$P=12$$ρ$$169vt2$$−$$vt2=12$$ρ$$7vt29$$∴$$N=9$$

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