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(D) Consider the frame of reference of the train. In this frame,the train is at rest,and the tunnel moves with speed $v_t$. The air in front of the tr

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$9$

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(D) Consider the frame of reference of the train. In this frame,the train is at rest,and the tunnel moves with speed $v_t$. The air in front of the train moves towards the train with speed $v_t$.Let $v$ be the speed of the air in the gap between the train and the tunnel walls relative to the train. The cross-sectional area of the gap is $A_{gap} = S_0 - S_t = 4S_t - S_t = 3S_t$.Applying the equation of continuity for the air flow relative to the train:$S_0 v_t = A_{gap} v$$4S_t v_t = 3S_t v$$v = \frac{4}{3} v_t$Now,apply Bernoulli's equation for the air flow along a streamline from the front of the train to the gap region:$p_0 + \frac{1}{2} \rho v_t^2 = p + \frac{1}{2} \rho v^2$$p_0 - p = \frac{1}{2} \rho (v^2 - v_t^2)$Substitute $v = \frac{4}{3} v_t$ into the equation:$p_0 - p = \frac{1}{2} \rho \left( (\frac{4}{3} v_t)^2 - v_t^2 \right)$$p_0 - p = \frac{1}{2} \rho (\frac{16}{9} v_t^2 - v_t^2)$$p_0 - p = \frac{1}{2} \rho (\frac{7}{9} v_t^2) = \frac{7}{18} \rho v_t^2$Comparing this with the given expression $p_0 - p = \frac{7}{2N} \rho v_t^2$:$\frac{7}{2N} = \frac{7}{18}$$2N = 18$$N = 9$

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