First slide

Instantaneous velocity

Question

The displacement x of a particle varies with time t as $x = a e^{- α t} + b e^{β t}$ , where a, b, $α$ and $β$ are positive
constants. The velocity of the particle will

Moderate

A

go on decreasing with time
B

be independent of $α$ and $β$
C

drop to zero when $α = β$
D

go on increasing with time

Solution

Given, $x = a e^{- α t} + b e^{β t}$
So, velocity, $v = \frac{d x}{d t} = - a α e^{- α t} + b β e^{β t}$
$a = \frac{d v}{d t} = a α^{2} e^{- α t} + b β^{2} e^{β t} = + ve all time.$
$∴$ v will go on increasing.

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