Terminal velocity-a "falling" miracle

A brief introduction to terminal velocity

Free-falling through the air without a parachute would be a terrifying thought to many people, but what about falling in water? Honey? Condensed Water Vapour? Why do we fall faster in some environments or fluids, while we fall slower in others or perhaps even stop falling? The phenomenon of things falling ultimately ties to a concept known as terminal velocity.

Before we move onto terminal velocity, we need to understand some basics first. Things fall from a height to the ground because of the Earth’s gravity attracts them towards the Earth’s centre. We can only fall through fluids (liquids and gas) since solids have their particles and molecules arranged in a strict shape and do not allow anything to pass through. When falling, we experience something called drag. Drag is an opposing force that exists when you move against the particles in the fluid. When there’s more particles to shove out of your way for you to move, there is more drag. For example, let’s say you throw a punch in the air. Next, you throw a punch while in the swimming pool. Why is your punch in the air faster than the one in water? Because there is less drag in the air. Since there are less molecules in the air at a single spot than in the water, your fist will move faster and experience less drag.

When things are moving, they all have a net force, which is represented by Fnet=ma, from the famous 2nd Law of Newton’s Laws of Motion. Net force is the sum of forces in all directions, and tells how much force is on the object, and in which directions. To make things simple, let’s say you’re falling through the air straight down. You have the force of gravity pulling you down, while the drag force is cancelling part of gravity’s force on you.

Note: if you’re in a liquid such as water, there’s also a buoyant force acting upwards so you’ll have to factor that into your equation as well! But for simplicity’s sake, let’s focus on a gas like air for now and later in the equation solving.

You might ask: how does terminal velocity come into play with this? Well, as stated in the previous paragraph, drag force cancels out part of gravity’s force on you. So what if instead of cancelling out part of it, it cancels the force of gravity out completely? When two opposing forces in the same dimensional axis cancel each other out (such as normal force and gravity on you right now), the net force becomes zero. No net force means no acceleration, and no acceleration means no change in velocity. Which is why when a falling object’s net force reaches zero, they reach their terminal velocity, the highest speed in a direction that they’ll ever achieve in that fluid.

However! This does not mean when your drag force and your force of gravity cancel each other out, you stop moving completely. You might stop in some situations, but remember it’s your acceleration that ceased to exist, not your velocity. If you were falling at 15 m/s as you hit terminal velocity, you’ll continue moving at 15 m/s in that fluid and in that direction from that point onwards. Zero net force means no change in velocity, not zero velocity.

Now that we’ve understood how terminal velocity works, let’s take a look at how to find it through equations.

Looking at the diagram above, we have two forces acting on our falling object: the force of gravity and the drag force. You probably have already heard about the force of gravity, but just in case you hadn’t: The formula for the force of gravity is Fg = mg, where m is the mass of said object, and g is the gravitational acceleration of the planet the object is on.

Gravity is an easy force to figure out how to solve, but Fd, the drag force, is trickier.

The formula for the drag force is Fd = ½ρv2CdA. There are a lot of terms here, so I’ll break it down:

ρ (rho, not p) is the density of the fluid that the object is travelling through. If an object is denser, there’s more things to push out of the way in a certain area, which increases the drag force that the fluid acts on the object as it falls/moves through it. 

v is the velocity of the object relative to the fluid while it’s moving in it. Relative velocity of an object to a fluid = the velocity of an object relative to Earth - the velocity of the fluid relative to Earth (vO/F = vO/E - vF/E). For example, if the object is moving at 10 m/s (relative to Earth) , but the fluid is moving at 5 m/s in the same direction (relative to Earth), then the object is moving 5 m/s relative to the fluid.

A is the cross-sectional area. In simple terms, “The cross-sectional area is the area of a two-dimensional shape that is obtained when a three-dimensional object - such as a cylinder - is sliced perpendicular to some specified axis at a point. For example, the cross-section of a cylinder - when sliced parallel to its base - is a circle”, as said by www.energyeducation.ca. 

Finally, Cd is the drag coefficient. The drag coefficient is a dimensionless number used to quantify the resistance against movement of an object in a fluid. Although the drag coefficient usually is found by the fluid the object’s in, it can also have something to do with the shape of the object travelling through the fluid. There is an equation for the drag coefficient, which is Cd = 2Fd / ρv2A, but it’s basically just the drag force equation rearranged. The only difference is that the v in this equation is a scalar quantity, speed, and A is the reference area. The reference area is the front of the object that is in direct contact with the fluid as it travels in a direction. Here’s a chart on some of the shapes of objects and their drag coefficient:

Now that we worked out how to find the drag force and the drag coefficient, let’s get back to our main concern: terminal velocity. As I’ve said 8-9 paragraphs ago, terminal velocity is reached when your drag force and force of gravity manage to cancel each other out, and your net force results in zero. The net force of an object going downwards is W (the object’s weight) minus the drag force, Fd.

So: Fnet = W - Fd. But since we are reaching terminal velocity in this situation, drag force and force of gravity cancel each other out by equalling each other in magnitude and in opposite directions. Which means Fnet = 0, and W - Fd = 0, leading to W = Fd.

Now we plug in the equation for the drag force: W = ½ρv2CdA

Then rearrange the equation to isolate v2: v2 = 2W/ρCdA, before square rooting both sides…

And voila! We have our terminal velocity equation, v = √(2W/ρCdA). This is used to solve how fast an object could possibly reach while it’s travelling through a fluid, and extends to many applications beyond finding just how fast the object moves. For example, the results can be applied in order to study the settling of sediments near the ocean bottom and the fall of moisture drops in the atmosphere. This principle is also applied in the falling sphere viscometer, an experimental device used to measure the viscosity of highly viscous fluids, such as oil and tar.

In conclusion, terminal velocity gives us a clear sense of our motion’s limits in a specific environment, and through its extended applications, we could figure out much more regarding safety and other factors to consider in experiments we perform. The concept itself may seem to be but a tiny part of the grand unification of physics, but every trinket of the whole may surprise you on how many other laws and theories it might support.