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Question

# If velocity v of a particle as a function of position x is given as v=(α−βx) where at t=0,x=0. Then,

A
Maximum displacement of particle is αβ.
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B
Maximum displacement of particle is .
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C
It takes time for particle to stop.
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D
Time taken to reduce to half of initial velocity is ln(2)β.
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Solution

## The correct option is D Time taken to reduce to half of initial velocity is ln(2)β.v=α−βx⇒dxdt=α−βx⇒∫x0dxα−βx=∫t0dt x=αβ(1−e−βt). ∴ The maximum value of x is αβ, and it happens at t=∞. Since, displacement is maximum at t=∞, its velocity should be zero. v=dxdt=−αe−βtv0=−α Say at t=t1/2,v=v0/2 ⇒v02=−αe−βt1/2⇒v02=−v0e−βt1/2⇒t1/2=ln2β

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