Terminal Velocity

Bookmark added to your notes.
View Notes
×

What is Terminal Velocity?

Let's say you are on a tall building with a ball. You drop the ball and it accelerates with 9.81 m/s2 . But at the same time the ball is being pushed backwards or decelerated by the air resistance. As the velocity of the ball increases, the air resistance increases. This happens till the air resistance deceleration equals the gravitational pull acceleration. This is when the ball reaches terminal velocity as there is no net acceleration on the ball.

[Image will be uploaded soon] [Image will be uploaded soon] [Image will be uploaded soon]

Gravitational Pull only                       Small Air Resistance               Gravitational Pull = Air resistance

Terminal velocity is the maximum attainable velocity of an object falling through a liquid or a gas. This happens when the resistance offered by the fluid increases with the increase in the magnitude of velocity. Eventually the object reaches terminal velocity as soon as the resistance becomes equal to the driving force.


Drag Force

It is a resistive force which acts opposite to the relative motion of an object with respect to a fluid. It is different from other resistive forces in the sense that it depends on the velocity of the object falling through the fluid. It is commonly referred to as air resistance or fluid resistance. Drag coefficient is a quantity that quantifies all the complexities of dependencies of resistive force and is dimensionless.


Drag Equation

Drag force depends on the density, compressibility and the viscosity of the fluid , the square of the velocity of the object , the size and shape of the body, and inclination of the body towards the flow .Generally ,the dependence on other properties is complex to show Mathematically.

We can  deal with these complex dependencies by  characterizing them in a single variable. For drag, this variable is called the drag coefficient, designated Cd.The value of drag coefficient can be calculated by experimentation. 

The drag equation states that drag D is equal to half of the density r times drag coefficient Cd times multiplied by the square of the velocity v of the object times the projected area.

D = 1/2ρv2ACd

Mathematical Representation of Terminal Velocity

VT = √2mg ∕ρACd

where

  • VT represents terminal velocity

  • m is the mass of the falling body

  • g is the acceleration due to gravity 

  • A is the projected area of the object

  • ρ is the density of the fluid

  • Cd represents the drag coefficient


Derivation of Terminal Velocity

Drag equation gives-

D = 1/2ρv2ACd

Net force exerted on the body-

Fnet = ma = Gravitational force - Drag force

ma = mg - 1/2ρv2ACd

At equilibrium, F = 0 this implies

0 = mg - 1/2ρv2ACd

mg = 1/2ρv2ACd

By solving this,

VT = √2mg ∕ρACd


Graphical Representation

The example we discussed above can be represented graphically.

The velocity of the falling ball is increasing under the influence of the gravitational pull but the air resistance opposes this increase. Thus, at some point, both the forces cancel out and the ball attains maximum possible velocity.The velocity of the falling ball remains constant from that point

[Image will be uploaded soon]

 

Did you know?

  • Drag force of a fluid is ultimately a cause of the viscous force but surprisingly its Mathematical representation is independent of the viscosity of a fluid.

  • Buoyant force is also a type of resistive force which opposes the motion of a body in a fluid.

  • Evolution has forced birds to be streamlined to fly better to reduce drag and similarly fish to swim faster underwater.

  • Aerodynamic drag is responsible for the balance of aircrafts in the sky.

  • Terminal velocity of a human in air is 150m/s so if we fall on a water body with this much speed, there will be massive collateral damage.

  • A ball will have different terminal velocities on different planets due to change in their gravitational effect.

FAQ (Frequently Asked Questions)

1. How does the shape of the body impact its Terminal velocity?

Ans. Shape of the body highly impacts the magnitude of the terminal velocity. Although the mathematical relation is very complex, we can see that spherical shapes have more terminal velocities due to their area - volume ratios. Broader shapes tend to cause more air resistance and hence skydivers are trained to use the flaps to tackle air resistance. There are many other applications when shape is modified to either increase or decrease the terminal velocity.

2. What are some of the ways to reduce drag on the  object falling through a fluid?

Ans. Golf balls have dimples created on them in order to spear through the air. Similarly the clothes of professional cyclists are being redesigned to reduce the drag force. New types of swimming suits are designed so as to be more streamline and be faster inside water. All these applications are basically derived by making the bodies in such a way that they experience low values of drag.

3. Why can a squirrel jump from a tree branch to the ground and escape undamaged, while a human could break a bone in such cases?

Ans. A squirrel can practically execute these jumps due to its size. Squirrels have a very large surface area in comparison to their mass. Drag force is proportional to the surface area. Thus, the drag force opposing the gravitational pull is quite large and hence the squirrel remains unchanged. On the other hand, a human has a lot of mass and in turn greater gravitational force acts on it which could not be sufficiently opposed by the drag force. Therefore, the impact may break a bone or two in this case.