In physics, “Terminal Velocity” (also referred to in the literature as limit speed, boundary velocity, or final velocity) is the constant speed that an object reaches when falling through a fluid medium (typically air), where the force of gravity is balanced by air resistance (drag force). When an object begins to fall, it accelerates; however, as its speed increases, air resistance also increases. At a certain point, this resistance becomes equal to the object's weight, and the net force becomes zero. From this moment on, the object continues to fall at a constant speed, which is defined as the terminal velocity.

Mathematical Expression of Terminal Velocity

Terminal velocity is the constant speed reached when the object's weight is balanced by the drag force. Based on Newton's Second Law and the general expression of drag force, terminal velocity is defined as follows:

$v_t = \sqrt{\frac{2mg}{\rho A C_D}}$

• Vt: Terminal velocity (m/s)
• m: Mass of the object (kg)
• g: Gravitational acceleration (approximately 9.81 m/s²)
• ρ: Density of the fluid (e.g., air) (kg/m³)
• A: Cross-sectional area of the object in the direction of motion (m²)
• CD: Drag coefficient (depends on the shape and size of the object, dimensionless)

Sample Problem

A skydiver with a mass of 70 kg is released under conditions where the air density is 1.2 kg/m³, the vertical cross-sectional area is 0.7 m², and the drag coefficient is 1.0. Given that the gravitational acceleration is 9.81 m/s², calculate the terminal velocity the skydiver will reach due to the effect of air resistance.

$v_t = \sqrt{\frac{2mg}{\rho C_D A}} \Rightarrow v_t = \sqrt{\frac{2 \times 70 \times 9.81}{1.2 \times 1.0 \times 0.7}} \Rightarrow v_t = \sqrt{\frac{1373.4}{0.84}} = \sqrt{1633.57} \approx 40.43\, \mathrm{m/s}$

The skydiver’s terminal velocity will be approximately 40.4 m/s.

The Importance of Terminal Velocity: A Study on Falling Raindrops

Terminal velocity is the constant speed attained by a freely falling object when the force of gravity is balanced by air resistance. This concept is of critical importance for understanding the dynamic behavior of moving objects.

Using raindrops as an example, it can be observed that terminal velocity acts as a natural speed limit. If there were no opposing force limiting the falling speed of raindrops—i.e., if the concept of terminal velocity didn’t exist—the drops would continuously accelerate under the influence of gravity and reach much higher speeds. This would increase the impact force of the raindrops hitting the ground, potentially leading to harmful effects.

However, air resistance in the atmosphere limits the speed of falling raindrops and ensures they reach a constant terminal velocity. This keeps their kinetic energy at a manageable level, minimizing the damage they might cause upon hitting the ground. The regulating effect of terminal velocity serves as a protective mechanism in many natural free-fall events.

Thus, terminal velocity is not only a physical phenomenon but also plays a vital role in stabilizing natural processes and protecting living environments.