Motion under influence of a resistive force

[Ananthan9470]

Consider the 1d motion of a body under the influence of the force given by F = -m*γ*v^α. m is mass, γ is a constant of appropriate dimension, v is velocity and α is dimensionless constant. The value of α for which the motion will come to a stop in finite time is to be calculated. I solved the equation of motion given by v'' = -γ*v^α and got an equation for t given by t = (1/γ)*(V₀^1-α)/(1-α); V₀ is the initial velocity. According to this, for all γ>1, time is finite. But the sign of t is negative. Why is this happening? Am I doing something wrong?

 

[Chestermiller]

Your solution for t is correct. What do you get for t if $\alpha = 0$? Is this a finite time? Is it positive?

 

[Ananthan9470]

Your solution for t is correct. What do you get for t if $\alpha = 0$? Is this a finite time? Is it positive?

For $\alpha = 0$ it is positive but what about something like α = 2? It is turning out to be negative unless I am mistaken. right?

 

[Chestermiller]

For $\alpha = 0$ it is positive but what about something like α = 2? It is turning out to be negative unless I am mistaken. right?

Yes. That's the region that is not physically realistic.

What do you get when $\alpha = -1$? Is that a finite time? What do you get for v vs t in the special case when $\alpha = 1$?