What is Terminal Velocity?
The maximum speed or velocity that an object can attain as it falls through the fluid is called terminal velocity. It happens when the sum of drag force from the fluid and buoyancy is equal to downward gravitational force acting on the object. In these situations, the net force that acts on the object is 0. This is the reason why the object has 0 acceleration.
In fluid dynamics, the object is moving at terminal velocity if the object speed is constant. The reason behind speed being constant is the restraining force which is exerted on an object by fluid via which that specific object is moving. It is possible to change the terminal velocity of an object by the fluid properties, object’s predicted cross-sectional surface area, and mass of the object. Let us have a look at the Mathematical Representation of the terminal velocity and derivation of the terminal velocity.
The Mathematical Representation of the Terminal Velocity
Using Mathematical terms, the formula for derivation of terminal velocity without the buoyancy effects can be described as:
Vt = \[\sqrt{\frac{2mg}{pACd}}\]
Here, Vt depicts terminal velocity, ‘mis’ represents mass of falling object, ‘g’ is the acceleration of the object because of velocity, ‘Cd’ is the drag coefficient, ‘p’ stands for fluid density via which the object is falling, and ‘A’ is the protection of object. In reality it is considered that the object can attain terminal velocity asymptomatically. The effect of buoyancy is upward force which is acting on the object from the surrounding field. The effect of buoyancy can be considered within the formula using Archimedes principle.In the equation, the mass m needs to be reduced by displaced fluid’s mass, that is pV, where V is the object volume. After that, take Mr = m - pV instead of m in all of the subsequent formulas.
Derivation of Terminal Velocity
The solution for answering the question ‘How to derive the expression for terminal velocity?’ is provided in the Vedantu notes on “Terminal Velocity Derivation”. The derivation of terminal velocity using Mathematical terms according to the drag equation comprehensive stepwise equations to arrive at the desired equation. First we use the formula for the free fall of the object then we derive the equation assuming that free fall happens in the positive direction. Next we derive the differential form of the equation, and then we integrate the equations. The complete stepwise derivation is available with Vedantu notes that covers the different aspects related to “Terminal Velocity Derivation”.
Terminal Velocity in Presence of Buoyancy Force
When the effects of buoyancy are considered, then the object that is falling through the fluid beneath its own weight can possibly reach the terminal velocity, if net force that is acting on the specific object becomes 0. When this object has obtained terminal velocity, the object weight gets balanced by the force of upward buoyancy, and the drag force. This is represented by,
W = Fb + D
Here, W is object weight, Fb is buoyancy force, and D is drag force which acts on the object.
If the falling object has spherical shape, then expressions for the forces (W, Fb, D) are as follows.
W = \[\left ( \frac{\pi }{6} \right )d^{3}\] PsG,
Fb = \[\left ( \frac{\pi }{6} \right )d^{3}\] PG
D = Cd(½) pV \[^{2}\] A
How is Terminal Velocity Useful?
When an object is dropped from rest, its speed will increase until it reaches terminal velocity. When an object is forced to move faster then its terminal velocity will slow down and reach to its constant velocity upon release.Terminal velocity occurs when the speed of a moving object is no longer increasing or decreasing and the object’s acceleration or deceleration is zero.
Objects Fall at Different Speeds- Why?
When the object falls down, the speed of the falling object increases continuously. As the speed of the falling object gets faster and faster, the air drag force increases and it is equal to the force of gravity. Thus, there is no net force acting on the object. When two forces acting on an object are balanced, the object will neither speed up or slow down but it will continue falling at a constant velocity and this is called the terminal velocity. The air drag force depends on the size and shape of the object. Objects having a large surface area such as in a parachute will have a much lower terminal velocity than objects with a smaller surface area such as a person falling from a plane. The weight of an object also affects the air drag force on an object and its terminal velocity. This also explains why a crumpled piece of paper will fall faster than a flat piece of paper. The weight of the paper is the same but the air drag force reduces because the surface area and drag coefficient change. This produces higher terminal velocity of the crumpled paper than the flat paper.
FAQs on Terminal Velocity Derivation
1. What is terminal velocity?
Terminal velocity is the constant maximum speed that a freely falling object eventually reaches when the resistance of the medium (such as air or water) through which it is moving equals the force of gravity. At this point, the net force acting on the object is zero, and its acceleration becomes zero, so it no longer speeds up.
2. How is the formula for the terminal velocity of a sphere falling through a fluid derived?
The derivation involves balancing the forces acting on the sphere when it moves at a constant velocity. The steps are as follows:
- Identify the forces: Three forces act on the sphere: the downward gravitational force (Weight, W), the upward buoyant force (Fb), and the upward viscous drag force (Fd).
- Establish Equilibrium: At terminal velocity, the net force is zero. Therefore, the downward force equals the sum of the upward forces: W = Fb + Fd.
- Substitute Formulas: For a sphere of radius 'r', density 'ρ', falling through a fluid of density 'σ' and viscosity 'η', we substitute the expressions for each force.
- Weight (W) = (4/3)πr³ρg
- Buoyant Force (Fb) = (4/3)πr³σg
- Drag Force (Fd) = 6πηrvt (from Stokes' Law)
- Solve for Terminal Velocity (vt): By rearranging the equilibrium equation and solving for vt, we arrive at the final expression for terminal velocity.
3. What are the main forces that enable an object to achieve terminal velocity?
Three primary forces are involved in achieving terminal velocity:
- Gravitational Force (Weight): The constant downward pull of gravity on the object's mass.
- Buoyant Force: The upward force exerted by the fluid, as described by Archimedes' principle. It is equal to the weight of the fluid displaced by the object.
- Viscous Drag Force: The resistive force exerted by the fluid that opposes the object's motion. This force increases as the object's speed increases.
Terminal velocity is reached when the gravitational force is perfectly balanced by the sum of the buoyant and drag forces.
4. What are the key factors that influence an object's terminal velocity?
An object's terminal velocity is primarily influenced by the following factors:
- Mass and Shape of the Object: A heavier or more streamlined object generally has a higher terminal velocity.
- Cross-sectional Area: A larger area perpendicular to the direction of motion increases drag, which lowers the terminal velocity (e.g., an open parachute).
- Density of the Fluid: A denser fluid exerts greater buoyant and drag forces, which typically results in a lower terminal velocity.
- Viscosity of the Fluid: A more viscous fluid offers more resistance, leading to a lower terminal velocity.
5. Why does an object stop accelerating once it reaches terminal velocity?
An object stops accelerating because the net force acting on it becomes zero. According to Newton's Second Law (F_net = ma), acceleration ('a') is directly proportional to the net force ('F_net'). When an object first starts to fall, the drag force is small, and the net downward force causes it to accelerate. As its speed increases, the upward drag force also increases. Terminal velocity is the exact point where the upward drag and buoyant forces perfectly cancel out the downward force of gravity. With F_net = 0, the acceleration must also be zero, causing the velocity to become constant.
6. What are some real-world examples where terminal velocity is important?
Terminal velocity is a crucial concept in many real-world situations, including:
- Skydivers: A skydiver reaches a terminal velocity of about 200 km/h. Opening the parachute dramatically increases the surface area and air resistance, lowering the terminal velocity to a safe speed for landing.
- Raindrops and Hail: Without air resistance, raindrops would hit the ground at extremely high speeds. Terminal velocity limits their speed, preventing them from causing significant damage.
- Sedimentation: In geology and chemistry, the rate at which particles settle in a fluid (sedimentation) is determined by their terminal velocity.
7. How does terminal velocity differ from the concept of free fall?
The key difference lies in the forces considered. Free fall is an idealised state of motion where the only force acting on an object is gravity, resulting in a constant downward acceleration (g ≈ 9.8 m/s²). In contrast, terminal velocity is a realistic scenario that occurs within a fluid (like air or water) and accounts for resistive forces like drag and buoyancy in addition to gravity. While an object in free fall continuously accelerates, an object reaching terminal velocity ceases to accelerate.
8. What happens if an object is forced to move through a fluid faster than its terminal velocity?
If an object is forced to move faster than its terminal velocity (for example, thrown downwards with great force), it will slow down until it reaches its terminal velocity. This occurs because at speeds above terminal velocity, the upward drag force becomes greater than the downward force of gravity. This creates a net upward force (or deceleration), which opposes the motion and reduces the object's speed until the forces balance out again at the terminal velocity.
9. Does terminal velocity exist in a vacuum? Explain why or why not.
No, terminal velocity does not exist in a vacuum. The concept of terminal velocity depends entirely on the presence of a resistive or drag force from a fluid medium (like air). In a perfect vacuum, there are no particles to cause air resistance. Therefore, the only force acting on a falling object is gravity, and it will continue to accelerate indefinitely without ever reaching a constant velocity. This is why a feather and a bowling ball fall at the same rate in a vacuum.
