Estimating Time to Reach Terminal Speed Using Drag & Unit Analysis

[Jstew]

Homework Statement

A marble of mass m falls through a fluid and is subject to the drag force F=-Av, where v is the velocity of the marble. The marble will reach a terminal speed given by v=(mg)/A. Use dimensional analysis to estimate how long it will take to reach the terminal speed. (Hint: A characteristic "time" can be constructed from A, g, and m.)

The Attempt at a Solution

The units on A are F/v= (kg/s).
The units on g are m/(s^2).
The units on m are kg.

However, g is the only value with meters, so I don't know how I'm supposed to combine them and get time, unless it's just m/A, but then why did the book give a hint about using g? Thanks for the help.

 

[Jstew]

/bump

 

[PhanthomJay]

Homework Statement

A marble of mass m falls through a fluid and is subject to the drag force F=-Av, where v is the velocity of the marble. The marble will reach a terminal speed given by v=(mg)/A. Use dimensional analysis to estimate how long it will take to reach the terminal speed. (Hint: A characteristic "time" can be constructed from A, g, and m.)

The Attempt at a Solution

The units on A are F/v= (kg/s).
The units on g are m/(s^2).
The units on m are kg.

However, g is the only value with meters, so I don't know how I'm supposed to combine them and get time, unless it's just m/A, but then why did the book give a hint about using g? Thanks for the help.

Yeah, the characteristic time, \tau, is, as you astutely noted, m/A. But this is derived from the terminal velocity (a function of m, g, and A) and the acceleration of gravity, g. So use dimensional analysis to find \tau by combining ,in some fashion, terminal velocity and g.