On October 14, 2012, Felix Baumgartner, a skydiver from Austria, did something incredible. He jumped from a height of 39,068 metres above Earth's surface, hoping to become the first skydiver to go faster than the speed of sound. Why did he jump from so high? It's because the law of gravity we learn about in school doesn't work the same way in our atmosphere. When something falls, it gets faster and faster because of gravity, but the air around it slows it down. This means there's a limit to how fast something can fall, called its "terminal speed" or "terminal velocity."

What is the definition of terminal velocity?

When an object moves through a medium that disperses energy, like air, it will eventually reach its maximum velocity. This is called its "terminal velocity" or "terminal speed." Sometimes we use the term "terminal speed" instead of "terminal velocity" because we already know the direction of movement. For example, if an object is falling towards the earth and affected by air friction, we know it's moving towards the earth and don't need to specify the direction.

"Terminal" means that the object can't go any faster than this speed, but it doesn't mean it will always reach it. Whether an object reaches its terminal velocity depends on the conditions of the system and the initial conditions of the object. Additionally, an object might start with a velocity greater than its terminal velocity, in which case it will eventually reach its terminal velocity from above instead of below.

What is the equation of terminal velocity?

Terminal velocity can't be calculated easily because it's an asymptotic quantity that depends on the specific details of each situation. While we can make estimates based on kinematics and energy, these approximations only work in some cases. Ultimately, we need to know the dynamics of the system at all times to accurately calculate terminal velocity.

Kinematic derivation

To understand how to calculate terminal velocity, we first need to consider the total force acting on a system. In the case of objects falling towards the earth, the total force is the gravitational force. The drag force, which captures the dissipative properties of the medium, opposes the movement of the object and tries to decrease its speed by dissipating its kinetic energy.

Newton's first law of motion tells us that an object's movement is not affected if there is no total force acting on it. Mathematically, we can equate the total force and the drag force to find the velocity at which the object will reach terminal velocity. However, to solve this equation, we need to know the exact form of the total force and the drag force, which requires a dynamical understanding of the system. In the next section, we will explore some examples of how to calculate terminal velocity explicitly.

Energetic considerations

While Newton's first law tells us that an unaffected system will not change, we can also derive the concept of terminal velocity from energetic considerations. In a dissipative medium, energy is lost as the body moves, reducing its speed and kinetic energy. This loss of energy prevents the body from reaching infinite speed under the influence of a force.

If we can model the loss of energy from the drag force, we can equate it to the gain of kinetic energy due to the force-generating movement. Newton's second law of motion, which relates forces to the change in momentum, can also help us understand the relationship between forces and kinetic energy. By combining these concepts, we can calculate terminal velocity for a given system.

Drag force and examples

When studying terminal velocity, the most common situation is a fluid slowing down the movement of bodies within it. The drag force formula for a fluid is given by ρv^2CD/2A, where ρ is the density of the fluid, v is the speed of the moving object, CD is the drag coefficient, and A is the area of the body perpendicular to the direction of movement. The drag force always opposes the movement of the body.

The formula shows that denser fluids oppose movement more strongly. The size and shape of the object are also quantified by the surface area and drag coefficient. The speed of the object determines the strength of the drag force, which leads to a stationary state of finite constant maximal velocity. This explains why there is a terminal velocity. By using these variables, we can calculate the terminal velocity of an object moving through a fluid.

Example of terminal velocity: an object falling towards the earth

For the gravitational force, we are using Newton’s law. We only consider the scalar value and not the direction of the force to simplify our computations. We consider a spherical body, such as a ball, falling.For spheres, the drag coefficient CD is 0.47. The density of the air is not constant, but we will take a situation close to the sea, where we have a value of ρ = 1.225 kg/m3. The area of a sphere perpendicular to the movement is that of a circle, that is, A = π·r2, and the mass and radius of the sphere are m = 1 kg and r = 1 metre.The gravitational force can be approximated by the formula:Here, g is the acceleration of the gravitational field close to the surface of the earth (9.81 m/s2), while h is the height over the surface of the earth. The drag force, with our data, has a value of:To compute the speed at all times, we would have to solve the system of equations using Newton’s second law, which involves solving differential equations. After having computed the general solution, we find that the value for the terminal velocity in this case is:

Example of terminal velocity: modified law of gravitation

Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object. Since the net force on the object is zero, the object has zero acceleration.[1] Terminal velocity can be calculated using the drag equation, which states that the net force acting on an object falling near the surface of Earth is (according to the drag equation): F net = m a = m g − 1 2 ρ v 2 A C d , {\displaystyle F_{\text{net}}=ma=mg-{\frac {1}{2}}\rho v^{2}AC_{d},} with v(t) the velocity of the object as a function of time t. At equilibrium, the net force is zero (Fnet = 0)[2] and the velocity becomes the terminal velocity limt→∞ v(t) = Vt: m g − 1 2 ρ V t 2 A C d = 0. {\displaystyle mg-{1 \over 2}\rho V_{t}^{2}AC_{d}=0.} Solving for Vt yields V t = 2 m g ρ A C d . {\displaystyle V_{t}={\sqrt {\frac {2mg}{\rho AC_{d}}}}.}[3]

The variables involved in calculating terminal velocity are mass (m), gravity (g), density (ρ), drag coefficient (Cd), and area (A). [1]

[1]: "Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object. Since the net force on the object is zero, the object has zero acceleration."
[2]: "At equilibrium, the net force is zero (Fnet = 0)"
[3]: "Solving for Vt yields V t = 2 m g ρ A C d ."

‍

Terminal Velocity - Key takeaways

Terminal velocity or terminal speed is the maximum velocity (speed) an object can reach while moving within a medium that dissipates energy (usually a fluid or gas). This occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object, resulting in a net force of zero and thus zero acceleration. The mechanism behind the appearance of a terminal speed can be traced back to the equilibrium between the energy gained by the body in its movement and the energy dissipated by the medium. The force that dissipates energy and that is always acting against the movement is called drag force. It has a widely known mathematical expression for fluids .

Using Newton’s first law, the terminal velocity can be calculated by solving for the drag force equation :

Fd = 1/2 ρCAv2

Where ρ is the density of the fluid, CA is the drag coefficient, and v is the velocity of the object.

Terminal Velocity

What is terminal velocity?

Terminal velocity is the maximum value of the speed an object can reach while moving within a medium that dissipates energy (usually a fluid or gas).

How do we calculate terminal velocity?

We can solve the dynamics either using Newton’s second law and study the asymptotic limit in time or using Newton’s first law and equate the total force to zero if the situation is mathematically simple enough.

How does weight affect terminal velocity?

For objects falling towards a body that attracts other bodies by gravitational interaction and that has a uniform fluid slowing them down, the terminal velocity depends linearly on the square root of the weight.

‍