# Terminal Velocity's Relationship to Mass

1. Oct 5, 2014

### bistan

1. The problem statement, all variables and given/known data
As shown in the plot above, terminal velocity is shown to increase linearly with the number of coffee filters dropped in a turbulent (air) medium. Therefore, terminal velocity depends on mass. Give an explanation for this starting from Newton's laws.

2. Relevant equations
$$\vec{F} = m\vec{a}$$
$$\vec{F}_{ab}=-\vec{F}_{ba}$$
Drag force $$F_{D}=\frac{1}{2}\rho v^{2}C_{D}A$$ where $$\rho$$ is the mass density of the fluid, $$v$$ is the velocity of the object relative to the fluid, $$A$$ is the reference (cross-sectional in our case) area of the object, and $$C_{D}$$ is the drag coefficient of the object in the fluid.

3. The attempt at a solution
By Newton's second law: $$\sum F=\sum ma=mg-F_{D}=mg-\frac{1}{2}\rho v_{T}^{2}C_{D}A$$
By definition of terminal velocity: $$\sum F=0 \Rightarrow 0=mg-\frac{1}{2}\rho v_{T}^{2}C_{D}A$$

This means that the sum of the drag force on the object and the weight of the object must be 0 for the object to have reached terminal velocity. Therefore an object with greater mass has a greater weight, and so the drag force on that object must be equally greater to bring the sum of the forces to 0.

Since the drag force is a function of the square of the terminal velocity on the object, the terminal velocity is greater for an object with greater drag force, and therefore is greater for an object with greater mass.

Last edited: Oct 5, 2014
2. Oct 5, 2014

### phinds

What does that statement say about the difference between the terminal velocity of a large styrofoam ball v.s. a small metal ball, assuming they have the same mass?

3. Oct 5, 2014

### bistan

If I understand correctly, I must also state that $$\rho, C_{D}, A$$ must also remain constant?