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A train of cross-sectional area Sₜ moves at speed vₜ through a long tunnel of cross-sectional area S0, where S0 = 4*Sₜ. Nearly all the air (density rho) ahead of the train flows back through the gap between the train's sides and the tunnel walls. The flow relative to the train is steady and laminar. The ambient pressure and the pressure inside the train are both p0. If the pressure in the gap region is p, and it is given that p0 - p = (7/(2N))*rho*vₜ², find N.
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Correct answer: 4
Solution
In the train frame, far air approaches at vₜ through area S0. The gap area is S0 - Sₜ = 3Sₜ. Continuity (all air goes through the gap): S0*vₜ = (S0 - Sₜ)*v_gap => 4Sₜ*vₜ = 3Sₜ*v_gap => v_gap = (4/3)vₜ. Bernoulli (ambient at vₜ, pressure p0; gap at v_gap, pressure p): p0 + (1/2)rho*vₜ² = p + (1/2)rho*v_gap². So p0 - p = (1/2)rho(v_gap² - vₜ²) = (1/2)rho((16/9 - 1)vₜ²) = (1/2)rho*(7/9)vₜ² = (7/18)rho*vₜ² = 7/(2*9) rho*vₜ². Hence N = 9. (Standard JEE 2020 answer is N = 9.)
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