18. The displacement x of a particle varies with time t as x = ae{−αt} +be{βt} , where a, b, α and β are positive constants. The velocity of the particle will

18. The displacement x of a particle varies with time t as \(x = ae^{−αt} +be^{βt}\)  , where a, b, α and β are positive constants. The velocity of the particle will

(a) be independent of α and β

(b) drop to zero when α = β

(c) go on decreasing with time

(d) go on increasing with time

Answer :  (d) go on increasing with time

\( x = a e^{-\alpha t} + b e^{\beta t} \)

\( v =\)\( \frac{dx}{dt} \)\(= - \alpha a e^{-\alpha t} + \beta b e^{\beta t} \)

As time increases

\(e^{\beta t}\) increases

\(e^{-\alpha t}\) decreases

so v will increase