OCR Specification focus:
‘Explain terminal velocity as constant speed when weight equals drag; interpret force and velocity graphs.’
When objects move through fluids like air or water, they encounter resistive forces that influence their motion. Terminal velocity describes the steady speed reached when these opposing forces balance perfectly.
Understanding Terminal Velocity
Terminal velocity is a fundamental concept in mechanics that applies to any object moving through a fluid under the influence of gravity. It occurs when the downward force of gravity is exactly balanced by the upward resistive force known as drag. At this point, the object ceases to accelerate and continues falling at a constant velocity.
Terminal Velocity: The constant speed reached by a falling object when the downward force of its weight equals the upward resistive (drag) force acting on it.
Once an object reaches terminal velocity, it moves at a steady speed because the net force on it is zero. Even though it continues to move, there is no further acceleration as dictated by Newton’s First Law of Motion.
Forces Acting on a Falling Object
When an object falls through a fluid (e.g. a skydiver through air or a ball through oil), several forces act upon it:
Weight (W): The constant gravitational force acting downward, calculated using W = mg.
Drag (D): The resistive force opposing motion, increasing with speed and dependent on fluid properties and object shape.
Resultant Force (F): The difference between weight and drag, determining acceleration according to Newton’s Second Law (F = ma).
At the start of the fall, drag is negligible, and acceleration is approximately equal to g (9.81 m/s²). As the object speeds up, drag increases until it equals the weight. Beyond this point, acceleration ceases, and terminal velocity is achieved.
Free-body diagrams for a skydiver show weight acting downward and drag upward. As speed increases, drag rises until it balances weight; the net force becomes zero and the motion continues at constant speed (terminal velocity). Numerical labels reinforce that acceleration falls to zero at equilibrium. Source
Evolving Motion Before Terminal Velocity
The motion of a falling object can be described in stages:
Initial Fall:
The object starts from rest.
Weight is much greater than drag, so acceleration is near g.
Increasing Speed:
As velocity increases, drag builds up significantly.
The resultant force (W – D) becomes smaller, reducing acceleration.
Reaching Equilibrium:
Eventually, drag equals weight.
Resultant force is zero, so acceleration stops.
The object continues at constant velocity — the terminal velocity.
Interpreting Force Graphs
Force–velocity graphs provide insight into how forces vary during the fall:
Initially, the drag force starts at zero and rises with velocity.
The weight line remains constant throughout.
The intersection point where the drag curve meets the weight line represents the terminal velocity — where net force equals zero.
Beyond this point, any further increase in velocity would increase drag beyond weight, creating a net upward force that would slow the object back down, maintaining equilibrium.
Evolving Drag with Velocity
Drag depends strongly on velocity. For many everyday objects moving through air, drag force (D) can be modelled as proportional to the square of velocity:
EQUATION
—-----------------------------------------------------------------
Drag Force (D) = ½ C ρ A v²
D = Drag force (N)
C = Drag coefficient (dimensionless, depends on shape and surface)
ρ = Fluid density (kg/m³)
A = Cross-sectional area (m²)
v = Velocity of the object (m/s)
—-----------------------------------------------------------------
As speed doubles, drag increases by a factor of four. This relationship explains why terminal velocity can be reached relatively quickly and why objects with large surface areas or irregular shapes experience lower terminal speeds.
Interpreting Velocity–Time Graphs
Velocity–time (v–t) graphs show how velocity changes with time when drag is present:
A velocity–time graph for a falling skydiver: the slope (acceleration) decreases as drag increases with speed. The graph levels off where drag equals weight, indicating terminal velocity. This directly supports interpreting the flattening region as a = 0. Source
Steep initial slope: High acceleration as drag is minimal.
Gradual flattening: As drag increases, acceleration decreases.
Horizontal section: Velocity becomes constant when terminal velocity is reached.
The area under a velocity–time graph represents the distance fallen, and this area continues to grow linearly once terminal velocity is achieved.
Factors Influencing Terminal Velocity
Several factors determine the terminal velocity of an object:
Mass (or Weight):
Heavier objects have greater gravitational force, requiring a larger drag force to balance it, leading to higher terminal velocity.Cross-sectional Area:
Larger areas encounter greater air resistance, reducing terminal velocity.Shape and Surface Texture:
Streamlined shapes (e.g. spheres, cones) reduce the drag coefficient (C), resulting in higher terminal velocities.Fluid Density:
Terminal velocity decreases in denser fluids because drag increases for the same velocity.Orientation and Motion Type:
The way an object moves through a fluid (tumbling vs streamlined descent) also alters drag and hence terminal velocity.
Everyday Examples
Skydivers are a classic example.
At the start, they accelerate under gravity with little drag.
As speed increases, air resistance builds until drag equals weight, producing a terminal velocity of around 55 m/s in a belly-down position.
By altering their body shape into a streamlined dive, they reduce drag and increase terminal velocity.
Similarly, raindrops and ball bearings in viscous liquids reach much lower terminal velocities due to smaller size and greater fluid resistance.
The Concept of Multiple Terminal Velocities
An object can reach different terminal velocities under varying conditions.
For instance, a skydiver who deploys a parachute experiences a sudden increase in drag due to the expanded surface area. The increased upward force causes deceleration until a new, much lower terminal velocity is established — safe for landing.
This demonstrates that terminal velocity is not a fixed property of the object alone, but a balance between forces that depends on the environment and orientation of motion.
Summary of Key Features
Terminal velocity is achieved when weight = drag.
At this point, acceleration = 0 and net force = 0, but motion continues at constant speed.
Force and velocity graphs visually represent this equilibrium point where forces balance.
Terminal velocity varies with mass, shape, area, and fluid density.
The principle applies universally to all objects moving through fluids, from falling rain to re-entering spacecraft.
FAQ
Terminal velocity is a vector quantity, meaning it has both magnitude and direction, while terminal speed refers only to the magnitude of that motion.
In vertical free fall, these two values are effectively the same because the direction is constant (downwards). However, in other situations—such as a projectile moving through a fluid—terminal velocity explicitly includes direction, which may be important when analysing motion in two or three dimensions.
Liquids such as oil or glycerine are much denser and more viscous than air. This increases the drag force for any given speed.
As a result, the drag grows to equal the object’s weight at a much lower velocity, meaning terminal velocity is reached more quickly and is significantly smaller.
Typical values:
Ball bearing in oil: terminal velocity of a few cm/s.
Skydiver in air: terminal velocity of around 50 m/s.
Temperature alters the viscosity and density of the fluid, both of which influence drag.
As temperature increases, most fluids become less viscous, reducing drag.
The reduction in drag allows an object to reach a higher terminal velocity.
Conversely, in colder conditions where viscosity increases, the same object will reach terminal velocity sooner and at a lower speed.
Yes, terminal velocity depends not only on mass but also on shape, surface texture, and cross-sectional area.
For example:
A smooth metal sphere and a rough ball of the same mass will fall at different terminal velocities because their drag coefficients differ.
The rougher or less streamlined the object, the greater the drag, resulting in a lower terminal velocity even when weight is identical.
Once terminal velocity is reached, the forces balance, so acceleration stops, but motion continues due to Newton’s First Law of Motion.
This law states that an object will continue moving at constant velocity unless acted on by an external unbalanced force.
In terminal motion, although there is no net force, the object’s inertia keeps it moving steadily. The drag continues to oppose motion, but its magnitude exactly balances the weight, maintaining equilibrium.
Practice Questions
Question 1 (2 marks)
A small steel ball is dropped into a tube of viscous oil. Describe what happens to the forces acting on the ball as it falls until it reaches terminal velocity.
Mark scheme:
(1 mark) Initially, the weight of the ball is greater than the drag, so the ball accelerates downward.
(1 mark) As speed increases, drag increases until it becomes equal to weight, and the ball then falls at constant velocity (terminal velocity).
Question 2 (5 marks)
A skydiver of mass 80 kg jumps from an aircraft and eventually reaches a terminal velocity of 55 m/s.
(a) Explain why the skydiver does not continue to accelerate indefinitely.
(b) Discuss two factors that could change the terminal velocity of the skydiver.
Mark scheme:
(a)
(1 mark) At first, weight acts downward with little air resistance, so the skydiver accelerates.
(1 mark) As velocity increases, air resistance (drag) increases.
(1 mark) When drag equals weight, the resultant force is zero, so acceleration stops and the skydiver falls at constant velocity (terminal velocity).
(b)
(1 mark) Body position or shape: A more streamlined shape reduces drag, increasing terminal velocity; a spread-eagle position increases drag, reducing terminal velocity.
(1 mark) Air density or surface area (e.g. parachute deployment): Greater air density or larger surface area increases drag, reducing terminal velocity.
