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The displacement x of a particle varies ...

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

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

Text Solution

Verified by Experts

The correct Answer is:
D
` 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.
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