JEE Advanced 2020 · Paper 2 · Physics Q2

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PhysicsInteger+3 / -1Question 2 of 54

A train with cross-sectional area $S_t$ is moving with speed $v_t$ inside a long tunnel of cross-sectional area $S_0\ (S_0 = 4S_t)$ . Assume that almost all the air (density $\rho$ ) 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 $p_0$ . If the pressure in the region between the sides of the train and the tunnel walls is $p$ , then $p_0 - p = \dfrac{7}{2N}\rho v_t^2$ . The value of $N$ is ______.

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Answer and worked solution

Official answer: 9

Work in the train's frame. Air approaches at $v_t$ through the full tunnel area $S_0$ , and almost all of it must pass through the gap of area $S_0-S_t=\frac34S_0$ .

Continuity.
$S_0v_t=\tfrac34S_0\,v\implies v=\tfrac43v_t.$

Bernoulli along a streamline, from far ahead (pressure $p_0$ , speed $v_t$ ) into the gap:
$p_0+\tfrac12\rho v_t^2=p+\tfrac12\rho\left(\tfrac43v_t\right)^2\implies p_0-p=\tfrac12\rho v_t^2\left(\tfrac{16}9-1\right)=\frac7{18}\rho v_t^2.$

Comparing with $\dfrac7{2N}\rho v_t^2$ gives $N=9$ .

Answer: 9

Topics:Bernoulli's principle Continuity equation

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JEE Advanced 2020 · Paper 2, Physics question 2