Theabsoultetemperature of airin a region linearly increases from T_1 to T_2 in a space of width d. Find thetime taken by a sound wave to go through the regionin terms of T_1, T_2d and the speed upsilon of sound at 273 K. Evalute this time for T_1 = 280 K, T_2 = 310 K, d 33m and upsilon = 330 m s^(-1)

Theabsoultetemperature of airin a region linearly increases from T_1 t

1.	Theabsoultetemperature of airin a region linearly increases from T_1 to T_2 in a space of width d. Find thetime taken by a sound wave to go through the regionin terms of T_1, T_2d and the speed upsilon of sound at 273 K. Evalute this time for T_1 = 280 K, T_2 = 310 K, d 33m and upsilon = 330 m s^(-1)
Answer» Solution :The VARIATION of TEMPERATURE is given by `T = T_1 + ((T_2-T_1))/(d) x …. (i)` We know that, `V prop sqrtT` `rArr (VT)/(V) = sqr((T)/(273))` `rArr VT = V sqrt((T)/(273))` `rArr DT = (dx)/(VT) = (du)/(V) XX ((273)/(T))` `t = (sqrt273)/(V) int_0^d (dx)/([T_1 + ((T_2 - T_1))/(d) x]` `= (sqrt273)/(V) xx (2d)/(T_2 - T_1) (sqrt barT_2 - sqrtT_1))` ` = ((2d)/(V)) ((sqrt273)/(T_2 - T_1)) sqrtT_2 - sqrtT_1` `T = (2d)/(V) (sqrt273)/(sqrtT_2 + sqrt T_1)` `[ Since T_1 - T_2 = (sqrtT_2 + sqrtT_1) (sqrtT_2 - sqrtT_1)]` Putting the given value we GET `(2xx33)/(330) . (sqrt273)/(sqrt280 + sqrt 310) = 96 ms`