Question

What is the equation for terminal velocity?

Answer 1

The terminal velocity of an object falling through a fluid (like air) can be determined by balancing the gravitational force acting on the object with the drag force acting against it.

The equation for terminal velocity \( v_t \) can be derived from the following forces:

1. Gravitational force:
\[
F_g = mg
\]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.

2. Drag force:
\[
F_d = \frac{1}{2} C_d \rho A v^2
\]
where \( C_d \) is the drag coefficient (depends on the shape of the object), \( \rho \) is the density of the fluid, \( A \) is the cross-sectional area of the object, and \( v \) is the velocity.

At terminal velocity, these two forces are equal:
\[
mg = \frac{1}{2} C_d \rho A v_t^2
\]

Rearranging the equation to solve for the terminal velocity \( v_t \):
\[
v_t = \sqrt{\frac{2mg}{C_d \rho A}}
\]

Thus, the terminal velocity can be expressed as:
\[
v_t = \sqrt{\frac{2mg}{C_d \rho A}}
\]
This equation indicates that terminal velocity increases with mass and decreases with drag coefficient, fluid density, and cross-sectional area.