Physics and Fluids: Accelerating Particles in a Fluid

[allezfou]

Homework Statement

Very small particles moving in fluids are known to experience a drag force proportional to speed. Consider a particle of net weight W dropped in a fluid. The particle experiences a drag force, Fd = kV, where V is the particle speed. Determine the time required for the particle to accelerate from rest to 95% of its terminal velocity, Vt, in terms of k, W, and g.

Homework Equations

Newtons Second Law of motion, etc.

The Attempt at a Solution

I tried to sum forces, etc. but didn't really get anywhere...

 

[Doc Al]

What forces act on the particle?

 

[allezfou]

i wrote the question verbatim, so i will assume a drag force (and a force of flowing fluid?)
i can write the answer if it helps, but it's useless without the method.

 

[Doc Al]

i wrote the question verbatim, so i will assume a drag force

Right. An expression for that is given.

(and a force of flowing fluid?)

That's the drag force.

What other force, also given, acts on the particle?

 

[allezfou]

there is a force on the particle moving it forward and the drag force

 

[Doc Al]

there is a force on the particle moving it forward

Yes. What is that force?

 

[allezfou]

i don't know.

 

[Doc Al]

i don't know.

Hint: It's one of the variables that your answer must be expressed in terms of.

 

[allezfou]

gravity.

 

[Doc Al]

gravity.

Of course! Now write an equation using Newton's 2nd law.

 

[allezfou]

so am i assuming a vertical pipe with fluid in it?

kV-mg=ma. we don't want it in terms of acceleration so we use a=dV/dt.

 

[Doc Al]

so am i assuming a vertical pipe with fluid in it?

It's just a particle placed in some fluid and allowed to fall.

kV-mg=ma. we don't want it in terms of acceleration so we use a=dV/dt.

Good. I would switch the signs around, so that "down" is positive (since you know it's going to fall down).

 

[allezfou]

mg-kV=m dV/dt. the net weight is W, which is also mg.
W-kV=m dV/dt

 

[Doc Al]

mg-kV=m dV/dt. the net weight is W, which is also mg.
W-kV=m dV/dt

Good. Now just rearrange and integrate.