A particle starts with velocity v0 at time t = 0 and is decelerated at a rate proportional to the square root of its speed at time t with constant of proportionality α. The total time for which it will move before coming to rest is

Question

A particle starts with velocity v0 at time t = 0 and is decelerated at a rate proportional to the square root of its speed at time t with constant of proportionality α. The total time for which it will move before coming to rest is (A) √v0 (B) (v03 / 2/α) (C) (2 v03 / 2/α) (D) (2 √v0/α)

Tardigrade Admin · Accepted Answer

u = v 0, a =-α √ v so, (d v/d t)=-α √v ⇒ ∫v00 (1/√v) d v=∫0t-α d t ⇒ t =(2 √v0/α)

Q. A particle starts with velocity $v_{0}$ at time $t = 0$ and is decelerated at a rate proportional to the square root of its speed at time t with constant of proportionality $α$ . The total time for which it will move before coming to rest is

$v_{0}$

$\frac{v _{0}^{3/2}}{α}$

$\frac{2 v _{0}^{3/2}}{α}$

$\frac{2 v _{0}}{α}$

$u = v_{0}, a = - α v$
so, $\frac{d v}{d t} = - α v$
$\Rightarrow \int_{v_{0}}^{0} \frac{1}{v} d v = \int_{0}^{t} - α d t$
$\Rightarrow t = \frac{2 v _{0}}{α}$

Q. A particle starts with velocity v0​ at time t=0 and is decelerated at a rate proportional to the square root of its speed at time t with constant of proportionality α. The total time for which it will move before coming to rest is

v0​​

αv03/2​​

α2v03/2​​

α2v0​​​