Velocity under a drag force

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Homework Statement


An object moving in a liquid experiences a linear drag force: D⃗ =(bv, direction opposite the motion), where b is a constant called the drag coefficient. For a sphere of radius R, the drag constant can be computed as b=6πηR, where η is the viscosity of the liquid.

Find an algebraic expression for vx(t), the x-component of velocity as a function of time, for a spherical particle of radius R and mass m that is shot horizontally with initial speed v0 through a liquid of viscosity η.
Express your answer in terms of the variables v0, η, R, t, m, and appropriate constants.

Homework Equations





The Attempt at a Solution


Thinking of typical dynamics, I divided the drag force, bv by m to get the acceleration. Then I subtracted acceleration times time from the inivial velocity, v0. So it looked like this:

v0- (6πηRv0t)/m.

Obviously it wasn't right, as my teacher today told me that I have to integrate the acceleration function to get the velocity function. I have no idea how to integrate the acceleration function in which it looks like every single variable are constants.

Help would be appreciated! Thanks in advance.
 

Answers and Replies

  • #2
tiny-tim
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hi playoff! :smile:
… I have no idea how to integrate the acceleration function in which it looks like every single variable are constants.
no, a = dv/dt is a function of v, not a constant :wink:
 
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Draw a free body diagram, and apply Newton's second law to the mass. Don't forget to include the buoyant force.

Chet
 
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hi playoff! :smile:


no, a = dv/dt is a function of v, not a constant :wink:
Ugh, I have a very shallow understanding in calculus. So if I would integrate it with v in the acceleration function, wouldn't it give me the position function in the velocity function? And the only variables I can use are v0, η, R, t, m, and appropriate constants.

Thanks for pointing it out though :D

@Chestermiller: I thought the only force acting in the x-axis is the drag force itself. Would the buoyant force also be acting against the velocity?
 
  • #5
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Ugh, I have a very shallow understanding in calculus. So if I would integrate it with v in the acceleration function, wouldn't it give me the position function in the velocity function? And the only variables I can use are v0, η, R, t, m, and appropriate constants.

Thanks for pointing it out though :D

@Chestermiller: I thought the only force acting in the x-axis is the drag force itself. Would the buoyant force also be acting against the velocity?
Oops. I should have read the problem statement more carefully. Sorry about that.

Chet
 
  • #6
tiny-tim
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(just got up :zzz:)
… the position function in the velocity function?
i don't understand this :confused:

to integrate dv/dt = f(v),

write it dv/f(v) = dt, then integrate both sides :smile:
 

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