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Question

The position x of a particle varies with time t as x=aeαt+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=aeαt+beβt, we get
v=dxdt=aαeαt+bβeβt
Differentiate the velocity to get the acceleration
dvdt=aα2eαt+bβ2eβt
dvdt=aα2eαt+bβ2eβt
From the equation above , it is easy to see that dvdt>0 for all t0 as the two positive terms are added. Thus , velocity v will keep on increasing with time.

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