Displacement of particle changes with respect to time according to equation `x =ae ^(-alpha t) + be ^(betat)` where a, b, a and Bare positive constants, then velocity of particle is ......

Options

A. indipendent of `alpha and beta`
B. will be zero if `alpha = beta.`
C. will increase with repsect to time.
D. will increase with respect to time.

Answer

D

Explanation

`x = ae ^(-alphat ) + be ^(beta t )`
`therefore (dx)/(dt) =a (e ^(- alpha t )) (-alpha ) + b ( e ^(beta r )) (beta)`
`therefore v =- aalpha e ^(- alpha t ) + b beta e ^(beta t )`
`= b beta e ^(beta t )- a alpha e ^(- alpha t)`
`= b beta e ^(beta t) - (a alpha)/(e ^(alpha t))`
Thus, `(a alpha )/(e ^(alpha t))` will increase with increase of t and velocity will increase.