The displacement x of a particle varies with time t as `x = ae^(-alpha t)+be^(beta t)`, where a, b,`alpha` and `beta` are positive constants. The velocity of the particle will

Options

A. Go on decreasing with time
B. Be independent of `alpha and beta `
C. Drop to zero when ` alpha =beta`
D. Go on increasing with time

Answer

D

Explanation

` x =ae^(-alpha t ) +be^(beta t) `
velocity ` v= (dx)/(dt) =(d)/(dt) (ae^(-alpha t)+be^(beta t) )= ae ^(-alpha t ) (-alpha ) +be^(beta t ) beta ) =-a alpha e^(-alpha t ) +b beta e^(beta t ) `
Acceleration ` =(dv)/(dt) =(d)/(dt) =-a alpha e^(-alpha t ) (-alpha ) +b beta e^(bt) beta = a alpha ^(2) e^(-alpha t )+ b beta ^(2) e ^(beta t ) `
Acceleration is positive so velocity goes on increasing with time.