The absoulte temperature of air in a region linearly increases from `T_1 to T_2` in a space of width d. Find the time taken by a sound wave to go through the region in terms of `T_1, T_2`d 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

A, B, C, D

Explanation

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`