On the 14th of October 2012, Felix Baumgartner, an Austrian skydiver, jumped from a height of 39,068 metres over the surface of the earth. His goal was to become the first skydiver to fall at a speed greater than the speed of sound. He had to jump from such a great height because Newton’s law of gravitation is not valid in the earth’s atmosphere: we have to take into account the contribution of air friction.
If there was no air friction, the acceleration caused by the gravitational field of the earth would continuously increase the speed of falling objects. However, when a force resisting the movement of objects appears, there is a maximum speed that objects can reach given their initial conditions: the terminal speed or terminal velocity.
What is the definition of terminal velocity?
Terminal velocity is the maximum velocity that can be reached by an object that is moving through a dissipative medium, that is, a medium that disperses energy.
In many settings, we find that terminal velocity can be substituted by the term ‘terminal speed’ because the direction of movement is fixed or known, and we only care about the magnitude of the speed. For instance, when we consider the terminal velocity of an object falling towards the earth that is affected by air friction, we know that the direction of the movement is radial towards the earth and hence do not need to specify it.
The adjective ‘terminal’ signals that the speed is the maximum speed that objects can reach, but it does not mean that objects will always reach this value. Whether they do depends on the properties of the system and the initial conditions of the object. In addition, while we may think that objects in a dissipative system will increase their velocity and slowly approximate the terminal value, they may also start with a velocity greater than the terminal one, in which case they will reach it from above rather than from below.
What is the equation of terminal velocity?
Terminal velocity is an asymptotic quantity that cannot be computed in a simple way because the details depend crucially on the details of the dynamics of each case. We can try to give kinematical and energetical approximations, which may apply to some situations, but we usually need the input of the dynamics (the evolution of the system at all times).
Kinematic derivation
First, let’s consider a certain force that is causing a system to move. We could consider complex combinations of forces, but here we will limit ourselves to the case where we have a total resulting force:
\[\vec{F}_T = \sum_i{\vec{F}_i}\]
Here, FT is the total force, while Fi are all the forces acting that are summed as vectors.
The resulting force in the case of objects falling towards the earth is the gravitational force. We now consider what is usually known as the drag force, which is nothing other than the resulting force that captures the dissipative properties of the medium. Since this force is dissipating energy, it ‘feeds’ on the kinetic energy of bodies and is thus opposed to their movement, trying to decrease their speeds.
The key to understanding how to compute the terminal velocity resides in Newton’s first law of motion. This law, which is valid for all physical systems, states that the movement of an object is not affected if there is not a total force acting on it. If the object was not moving, it would remain still, whereas if the object was moving, it would continue to do so at constant velocity.
Mathematically, this amounts to equating the total force acting on the body and the drag force, which allows us to solve for a certain value of the velocity. That is:
\[\vec{F}_T = \vec{F}_D\]
To solve this equation, we need the dynamical input of the exact form of the total force and the drag force. We will explore some examples of explicit calculations in the last section.
Energetic considerations
Considering Newton’s first law, it is clear by definition that when the total force equals zero, a final velocity will be reached. This is a possible situation when we have forces generating movement and forces opposed to it (drag force). Although the idea behind Newton’s first law is intuitive and logical (an unaffected system does not change), we will try to derive the concept of terminal velocity from purely energetic considerations.
A dissipative medium, as we already said, is a medium that disperses the energy of the body by reducing its speed, thus decreasing its kinetic energy. If a force generates a movement, an object under its influence will continuously accelerate and will, theoretically, reach an infinite speed at an infinite time. However, in a dissipative medium, this is not allowed to happen because there is a loss of energy towards the medium that prevents the kinetic energy from exceeding a certain value.
If we can model the loss of energy from the expression of the drag force, we only need to equate it to the gain of the kinetic energy of the body due to the force-generating movement. It turns out that some of these relations can also be found by using Newton’s second law of motion, which relates forces to the change of momentum, a quantity strongly related to the kinetic energy.
Drag force and examples
The most common situation where we study terminal velocity is for a fluid that slows down the movement of bodies within it. To properly study the examples below, we provide the experimental formula for the drag force of a fluid and briefly explain its dependence. The formula is:
\[F_D = \frac{1}{2} \cdot \rho \cdot v^2 \cdot C_D \cdot A\]
Here, ρ is the density of the fluid, v the speed of the moving object, CD a number called drag coefficient, and A the area of the body that is perpendicular to the direction of movement. The direction of this force is always opposed to the movement of the body.
As we can see from the mathematical formula, the denser the fluid, the bigger its opposition to the movement. The dependence of the formula on the characteristics of the object is quantified by three variables. Two of them correspond to their size (surface area) and shape (drag coefficient). The third corresponds to its state of movement and explains why there is a terminal velocity. The dependence on the speed indicates that the faster an object moves, the stronger the drag force is, which will lead to the aforementioned stationary state of finite constant maximal velocity.
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, \(A = \pi \cdot r^2\), 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:
\[F_g = m\cdot g \cdot h = 9.81 \cdot h \space N\]
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:
\[F_D = \frac {1}{2} \cdot \rho \cdot v^2 \cdot C_D \cdot A = 0.90 \cdot v^2 \space N\]
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:
\[v_T = \sqrt {\frac{2 m \cdot g}{\rho \cdot A \cdot C_D}} = \sqrt {\frac{19.62}{1.81}} = 3.30 (m/s)\]
Example of terminal velocity: modified law of gravitation
We consider a spherical body or 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, \(A = \pi \cdot r^2\), and the mass and radius of the sphere are m = 1 kg and r = 1 metre.
Imagine that the law of gravitation in our nature was different from the one we know as Newton’s law. Our proposed law also depends on g and on the mass of the falling object, but instead of depending linearly on the height, it depends linearly on the radial speed v:
\[\tilde{F}_g = m \cdot g \cdot v\]
Instead of having to solve the dynamics (which is a valid path to determine the speed, position, and acceleration at all times), we could consider Newton’s first law rather than the second one. The former states that the speed of an object will be constant when there is no force acting on it. Since we know that the terminal speed is constant, we know that the object will reach it when the forces have the same value.If we equate both forces, we obtain:
\[\tilde{F}_g = F_D \Rightarrow 9.81 \cdot v = 0.90 \cdot v^2 \Rightarrow v = 0 \text{ or } v = \frac{9.81}{0.9} = 10.9 \space m/s\]
These two solutions correspond:
- to the start of our experiment where there is no speed because we are dropping the ball
- and to the situation where a terminal speed has been reached.
Depending on the space available for the ball to fall, the speed reached will be the terminal velocity or a lower value (because maybe it did not have enough space to accelerate).
As we said before, had we released the ball with an initial speed greater than the terminal value, it would have evolved towards the terminal value, slowly decelerating until reaching it from above.
Figure 1. Evolution of speed with respect to time for a situation where terminal speed is reached from above (blue) and below (red). Usually, terminal velocity is reached asymptotically. Source: StudySmarter.
Terminal Velocity - Key takeaways
- Terminal velocity or terminal speed is the maximum value of the speed an object can reach while moving within a medium that dissipates energy (usually a fluid or gas).
- Usually, one needs to solve the dynamics to study the evolution of the position, velocity, and acceleration at all times in order to study the limit for long times. However, in some situations, one can use Newton’s first law to compute the value of the terminal speed.
- 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.
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