A particle moving in a straight line with initial velocity $u$ and retardation $α v$ , where $v$ is the velocity at any time $t$ .

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The particle will cover a total distance $\frac{u}{α}$ .

The particle will come to rest after time $\frac{1}{α}$ .

The particle will continue to move for a very long time.

The velocity of the particle will become $\frac{u}{2}$ after a time $\frac{1}{α}$ .

answer is A, C.

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Detailed Solution

Particle will not take U-turn.

$\begin{array}{l} v \frac{d v}{d x} = - α v or \int_{u}^{0} d v = - α \int_{0}^{x_{0}} d x or x_{0} = u / α . \\ \frac{d v}{d t} = - α v or \int_{u}^{v} \frac{d v}{v} = - α \int_{0}^{t} d t \\ or v = u e^{- α t} . ∴ v = 0 for t \to \infty . \end{array}$