Velocity of a Spherical Particle in a Viscous Liquid: Integrating Drag Force

[playoff]

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.

 

[tiny-tim]

hi playoff!

… 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

 

[Chestermiller]

Draw a free body diagram, and apply Newton's second law to the mass. Don't forget to include the buoyant force.

Chet

 

[playoff]

hi playoff!


no, a = dv/dt is a function of v, not a constant

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?

 

[Chestermiller]

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

 

[tiny-tim]

(just got up :zzz:)

… the position function in the velocity function?

i don't understand this

to integrate dv/dt = f(v),

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