14. starptautiskā konference 2012 - Latvijas Jūras akadēmija

Proceedings of 14th International conference „Maritime Transport and Infrastructure - 2012”Solution. The speed of radium disintegrationTherefore we can write an equation:dQ is proportional to amount of substance Q.dtdQdtwhere k is proportionality coefficient.After separation of variable and integration, we get: kQ , (11)dQQ kt;kt; ln Q kt ln C Q Ce(12)If t = 0, then Q = Q 0 , from (12) we arrive at C = Q 0 . Therefore:ktQ Q 0e . (13)ktln 21If t = t 1 , then Q = 0,5Q 0 , from (13) we get 0,5Q0 Q0eand calculate k .t1From (13) it follows:0tt ln 2t1 t1 Q0 2Q Q e. (14)Ift 10,1t , then from the formula (14) we get Q Q0e 0,9Q010.Problem 3. Sinking body. Body is slowly sinking in the water. Calculate draught s,depending on the time t, if at the starting point t = 0 is s = 0 and v = 0.Solution. Sinking speed v and acceleration a are mutually related by the formula:a = g – kv, (15)where k and g are constants.dsSince the speed is derivation of road by the time v , but the speed – derivation of speed by thedt2dv d stime a , by applying relation (15) we arrive at equation:2dt dt2d s ds k g.(16)2dt dtDifferential equation (16) is a linear non-homogenous differential equation with constant coefficients.So the solution can be expressed as a sum of respective homogenous differential equation’s2d s k2dtdsdt 0.(17)120