Q: Suppose that a body moves in a medium which offers drag to its motion, the drag force F being proportional to the velocity v of the body, raised to some power b, that is,  F = av[sup]b[/sup]
If the body is imparted an initial velocity, is it possible for it to cover an infinite distance?

Let us rephrase this question a bit. You have to figure out whether there is any value of b possible for which if the body is given an initial non-zero velocity, it never stops.

The question might seem strange. After all, if there is drag on the body, its velocity should be reduced continuously and finally become zero in some finite amount of time, so it should eventually stop. Well, there is no doubt about the truth of the statement “Its velocity should be reduced continuously”. But the second part, “The velocity finally becomes zero in some finite amount of time”, might not be very true. This is what you have to think about.
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sounds like a question for his/her undergraduate course?

if let a = 0
Fdrag= av[sup]b[/sup] =0

so Free Body Diagram apply Newton 2 law:

F = m*a
- Fdrag = m*a
0 = m*a
0 = a
It is a mathematical answer to achieve velocity to never decay by letting constant a =0,
i doubt it can be achieve in real life Earth bound conditions
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Hey! The drag is non-zero. That’s the whole point of this problem.

This means that a is not zero.