# The Naked Scientists Forum

### Author Topic: How does viscosity affect the time taken to reach terminal velocity?  (Read 4555 times)

#### mullermeister

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##### How does viscosity affect the time taken to reach terminal velocity?
« on: 05/11/2011 23:58:12 »
So if I have two liquid of different viscosity's, and I release identical ball bearings in each, the one in the lower viscosity liquid will ultimately reach a higher terminal velocity.
But how does the viscosity affect the acceleration?

I guess the higher viscosity will mean the ball accelerates slower, but if it reaches a lower terminal velocity, and travels slower, will it reach terminal velocity before the other bearing has travelled the same distance? (<-- that's what I'm really interested in)

Any thoughts? :)

#### Bored chemist

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##### How does viscosity affect the time taken to reach terminal velocity?
« Reply #1 on: 06/11/2011 10:41:42 »
In classical physics, nothing ever reaches terminal velocity.

#### JP

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##### How does viscosity affect the time taken to reach terminal velocity?
« Reply #2 on: 06/11/2011 15:06:41 »
Hi Mullermeister,

You have to make a lot of assumptions in order to answer your question easily.  The first is that viscosity's effect is a constant drag coefficient (meaning that the drag force is proportional to velocity and constant over all velocities).  This can be a good approximation, so long as the velocities are relatively small and the viscosity isn't too high.

Next, you have to tell us how to measure "reaching terminal velocity."  You never quite reach it, though you can get as close as you want.  The natural way to measure it is to say "how long will it take to reach X% of terminal velocity?"

Under those assumptions, the one with the higher drag is always going to win.

I know this because under the above assumptions this is fortunately a very simple problem to model in physics, and the result is:

v(t)=vterminal[1-exp(-t b/m)],

where t is time, m is the mass of the object and b is the coefficient of drag--b is larger when the drag is higher.  If you're math-savvy it's not too hard to see that from this equation, you reach any given % of terminal velocity faster when the drag is higher.
« Last Edit: 06/11/2011 15:13:28 by JP »

#### mullermeister

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##### How does viscosity affect the time taken to reach terminal velocity?
« Reply #3 on: 06/11/2011 15:12:29 »
Hi,

Thanks, I've got data from an experiment involving dropping bearings through different viscosity liquids, I was just trying to roughly work out whether they bearings in the higher viscosity liquid would 'reach' (this doesn't have to be really accurate) their terminal velocity after travelling a shorter distance than those in the lower viscosity liquid.