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.