The position x of a particle varies with time t as x=ae−αt+beβt, where a,b,α and β are positive constants. The velocity of the particle will
A
go on decreasing with time.
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B
be independent of α and β .
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C
drop to zero when α=β.
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D
go on increasing with time.
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Solution
The correct option is D go on increasing with time. Differentiate the displacement x=ae−αt+beβt, we get v=dxdt=−aαe−αt+bβeβt
Differentiate the velocity to get the acceleration dvdt=aα2e−αt+bβ2eβt ⇒dvdt=aα2eαt+bβ2eβt
From the equation above , it is easy to see that dvdt>0 for all t≥0 as the two positive terms are added. Thus , velocity v will keep on increasing with time.