Terminal Velocity Definition :

The maximum velocity that an object may go while falling through a fluid under the force of gravity such as air or water is known as its terminal velocity. When an object initially starts falling, it accelerates due to gravity. However, as it accelerates, it also experiences air resistance, or drag, which opposes its motion.

The force of air resistance increases in along with the object's velocity until it is at equal amount as the force of gravity pulling the object downward. At this point, the forces acting on the object balance each other out, causing the net force on the object to become zero. As a result, the object stops accelerating and falls at a constant velocity, known as terminal velocity.

Several factors affect terminal velocity. The object's mass, its shape, and the density of the air or water it's falling through. Bigger or less dense things, like a wide piece of paper, experience more air resistance, so they hit terminal velocity at lower speeds. Heavier or streamlined things, like a rock, need higher speeds to balance out gravity and air resistance.

Terminal Velocity Formula :

The formula for calculating terminal velocity involves the gravitational force acting on an object and the drag force due to air resistance. The equilibrium between these forces results in terminal velocity. The basic formula is expressed as:

Terminal Velocity (Vt) = √2mg / ρACd
Where :
  • (Vt) is the terminal velocity
  • m is the mass
  • g is the acceleration due to gravity
  • ρ is the density of the fluid
  • A is cross-sectional area of the object
  • Cd is the drag coefficient

This formula takes into account the forces acting on the object – gravitational force pulling it down and air resistance opposing its motion – and provides the speed at which these forces balance, resulting in a constant velocity during free fall.

Derivation of the Terminal Velocity Equation :

Balance of Forces Approach (Equilibrium of Forces):
Given:
  • Gravitational Force Fg = mg, where m is the mass of the object and g is acceleration due to gravity.
  • Drag Force Fd = ½ρv²ACd, where ρ is the density of the fluid, v is the velocity of the object, A is the cross-sectional area of the object facing the fluid and Cd is the drag coefficient.
Equilibrium of Forces at Terminal Velocity :

At terminal velocity, the net force acting on the object is zero because the object no longer accelerates. Therefore, we set the gravitational force equal to the drag force:
mg = ½ρv²ACd

Isolate Terminal Velocity v:

Rearranging the equation to solve for the terminal velocity v we get:
v² = 2mg / ρACd
Take the square root of both sides of the equation to solve for v:
v = √ 2mg / ρACd

This equation provides the terminal velocity of the object falling through the fluid. It depends on the mass of the object (m), the acceleration due to gravity (g), the fluid density (ρ), the cross-sectional area of the object (A), and the drag coefficient (Cd).

That's the step-by-step process for deriving the terminal velocity equation using the Balance of Forces Approach. It's based on the principle of equilibrium of forces at terminal velocity, where the gravitational force is balanced by the drag force.

How Terminal Velocity Calculator Works: A Step-by-Step Example

Question:

A spherical object of mass 0.76kg is falling through air with a drag coefficient Cd = 0.65, a cross-sectional area A = 0.12m² , and air density ρ = 1.4 kg/m³.What is its terminal velocity?

Answer:
To find the terminal velocity (vt) of the object, we use the formula:
(vt) = √2mg / ρACd
Given:
  • Mass of the object, m = 0.7 kg
  • Gravitational acceleration, g = 9.81 m/s²
  • Drag coefficient, Cd = 0.65
  • Cross-sectional area, A = 0.12m²
  • Air density, ρ = 1.4 kg/m³
Using the provided values:
(vt) = √2 × 0.7 × 9.81 / 1.4 × 0.12 × 0.65
(v t) = √13.734 / 0.1092
(v t) = √125.7692
(v t) = 11.214688 m/s
Therefore, the terminal velocity of the object falling through the air is approximately 11.214688 m/s.

Why to use our Terminal Velocity Calculator?

Using a terminal velocity calculator offers a quick and accurate means to determine the maximum velocity an object reaches when falling through a fluid, typically air or water. By inputting parameters such as mass, drag coefficient, cross-sectional area, and fluid density, the calculator employs relevant equations to compute the terminal velocity.

This tool eliminates the need for manual calculations, ensuring precision and efficiency in determining terminal velocities for various objects and conditions. Additionally, it aids engineers, scientists, and students in understanding the dynamics of objects in fluid environments, facilitating research, analysis, and problem-solving in fluid mechanics and related fields.