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Question: A particle moves with an initial velocity v<sub>0</sub> and retardation kv, where v is the velocity ...

A particle moves with an initial velocity v0 and retardation kv, where v is the velocity at any time t.

A

The particle will cover a total distance v0/k

B

The particle comes to rest at t = 1/k

C

Particle continues to move for a long time

D

At time 1/α, v = v0/2

Answer

Particle continues to move for a long time

Explanation

Solution

vdvdx=−kv\frac{vdv}{dx} = - kv or ∫v00dv=−k∫0xpdx\int_{v_{0}}^{0}{dv} = - k\int_{0}^{x_{p}}{dx}

or x0 = v0/k.

dvdt=−kv\frac{dv}{dt} = - kv or ∫v0vdvv=−∫0tkdt\int_{v_{0}}^{v}\frac{dv}{v} = - \int_{0}^{t}{kdt}

or v = v0e-kt ∴ v → 0 when t → ∞.