A train with cross-sectional area St is moving with speed vt inside a long tunnel of cross-sectional area S0( S0 = 4 St). 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=(7/2 N)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 = 4 St). 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=(7/2 N)ρ vt2. The value of N is. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-ua4z4tpqpwkfwa7w.jpg" /> A2=3St Equation of continuity vt4St=3Stv v=(4/3)vt Using Bernoulli's equation P0+(1/2)ρ vt2=P+(1/2)ρ ((4/3) vt)2 P0-P=(7/18)ρ vt2 Now compare from given value N=9

A2​=3St​ Equation of continuity vt​4St​=3St​v v=34​vt​ Using Bernoulli's equation P0​+21​ρvt2​=P+21​ρ(34​vt​)2 P0​−P=187​ρvt2​ Now compare from given value N=9

$A_{2} = 3 S_{t}$
Equation of continuity
$v_{t} 4 S_{t} = 3 S_{t} v$
$v = \frac{4}{3} v_{t}$
Using Bernoulli's equation
$P_{0} + \frac{1}{2} ρ v_{t}^{2} = P + \frac{1}{2} ρ (\frac{4}{3} v_{t})^{2}$
$P_{0} - P = \frac{7}{18} ρ v_{t}^{2}$
Now compare from given value
$N = 9$