Understanding Terminal Velocity

Usefulness in Physics and Fluid Dynamics

The concept of terminal velocity ($v_t$) is crucial in the study of mechanics, particularly in the fields of free fall and fluid dynamics. When an object falls through a fluid (like air or water), it accelerates due to gravity, but simultaneously experiences an increasing drag force proportional to the square of its velocity. Terminal velocity is the constant speed reached when the drag force equals the gravitational force (weight) acting on the object. At this point, the net force on the object is zero, and it stops accelerating.

This calculator is vital for applications like:

• Predicting the maximum speed of parachutists and skydivers.
• Analyzing the sinking speed of objects in liquids.
• Designing aerodynamic shapes for vehicles and projectiles to minimize drag.

Step-by-Step Instructions

1. Enter Mass ($m$): Input the mass of the falling object in kilograms (kg).
2. Enter Fluid Density ($\rho$): Input the density of the fluid (e.g., $1.225 , kg/m^3$ for air at sea level) in kilograms per cubic meter (kg/m³).
3. Enter Drag Coefficient ($C_d$): Input the dimensionless drag coefficient, which depends on the object’s shape (e.g., $\approx 0.5$ for a sphere, $\approx 1.0$ for a human skydiver).
4. Enter Cross-sectional Area ($A$): Input the maximum cross-sectional area of the object perpendicular to the direction of motion, in square meters (m²).
5. Calculate: Click the “Calculate Terminal Velocity ($v_t$)” button to get the result.

The Terminal Velocity Formula

The terminal velocity ($v_t$) is calculated using the following formula, derived from balancing the gravitational and drag forces:

Where the variables represent:

• $v_t$ = terminal velocity ($\text{m/s}$)
• $m$ = mass of the object ($\text{kg}$)
• $g$ = acceleration due to gravity ($9.81 \, \text{m/s}^2$)
• $\rho$ = density of the fluid ($\text{kg/m}^3$)
• $C_d$ = drag coefficient (dimensionless)
• $A$ = cross-sectional area ($\text{m}^2$)