OCR Specification focus:
‘Discuss factors affecting drag in air, including speed, cross-sectional area and surface characteristics.’
The factors affecting drag and terminal speed are essential to understanding how objects move through fluids, particularly air, where resistive forces determine acceleration, equilibrium, and motion stability.
Understanding Drag Forces
Drag is a resistive force experienced by an object moving through a fluid (a liquid or a gas). It always acts in the opposite direction to motion, reducing an object’s acceleration and eventually leading to a terminal velocity where acceleration ceases.
Drag: The resistive or frictional force that opposes the motion of an object through a fluid.
Drag arises from two main interactions:
Frictional contact between fluid particles and the object’s surface (viscous drag).
Pressure differences caused by the disturbance of fluid flow (pressure or form drag).
These components combine to form the total drag force, which is highly dependent on several factors, including speed, cross-sectional area, surface texture, and the nature of the fluid.
Relationship Between Drag and Speed
The dependence of drag on speed is one of its most critical characteristics. At low speeds, drag is often directly proportional to speed, but at higher speeds, the relationship becomes quadratic.
EQUATION
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Drag Force (Fₑ) = ½ × ρ × C_d × A × v²
Fₑ = Drag force (N)
ρ = Density of the fluid (kg m⁻³)
C_d = Coefficient of drag (dimensionless)
A = Cross-sectional area of the object (m²)
v = Speed of the object relative to the fluid (m s⁻¹)
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This equation demonstrates that drag increases rapidly with velocity. Doubling the speed results in a fourfold increase in drag. This explains why high-speed vehicles and projectiles require streamlined designs to reduce resistive effects.
Speed Regimes and Flow Types
Laminar flow: Smooth, ordered flow at low speeds. Drag is primarily viscous and increases linearly with speed.
Turbulent flow: Chaotic, eddy-filled motion at higher speeds. Drag depends on speed squared, producing significantly greater resistance.
The transition between laminar and turbulent flow occurs when the Reynolds number exceeds a certain threshold.
Laminar flow shows smooth, ordered streamlines with relatively low drag; turbulent flow shows chaotic eddies, a thicker boundary layer, and much higher drag at the same speed. The diagram highlights how regime changes underpin the steeper speed-dependence of drag. (Extra detail beyond the syllabus: the figure explicitly references Reynolds number thresholds.) Source
This transition explains why small objects falling slowly (like dust particles) experience negligible drag compared to faster, larger objects such as raindrops or parachutes.
Influence of Cross-Sectional Area
The cross-sectional area (A) represents the frontal area of an object facing the fluid flow. It determines how much of the fluid is displaced as the object moves, thus directly affecting drag.
A larger cross-sectional area increases the total drag force because more fluid molecules collide with the object per unit time.
A smaller cross-sectional area reduces drag, allowing faster or more energy-efficient motion.
Examples:
Skydivers extend their arms and legs to increase area and reduce terminal speed.
Vehicles, aircraft, and cycling helmets are designed with minimal frontal areas to reduce air resistance.
The quadratic relationship between drag and velocity means that reducing area is especially beneficial at higher speeds, where even small design changes can dramatically reduce resistive forces.
Role of Surface Characteristics
The surface texture of an object strongly influences the flow pattern of the surrounding fluid and hence the magnitude of drag.
Smooth Surfaces
Promote laminar flow, which reduces frictional drag at low speeds.
Are advantageous when movement occurs slowly or steadily through a viscous medium.
Example: Polished aircraft fuselages or racing car bodies minimise surface friction.
Rough Surfaces
Increase surface friction, but can also delay flow separation, thereby reducing pressure drag at high speeds.
Example: A golf ball has dimples that trip the boundary layer into turbulence, reducing wake size and total drag.
Airflow over a smooth ball separates early, producing a wide turbulent wake and higher pressure drag; a dimpled ball keeps the boundary layer energetic and attached longer, yielding a narrower wake and lower drag. This exemplifies how surface texture can reduce total drag despite added skin friction. (Note: the figure also mentions lift and spin context, which is ancillary to this syllabus point.) Source
Thus, the optimal surface texture depends on the speed regime and the dominant drag component. Engineers carefully balance these effects to design efficient vehicles and sports equipment.
Factors Affecting Terminal Speed
When an object falls through a fluid, drag increases with speed until it balances the weight of the object. At this point, net force = 0 and the object falls at a constant terminal velocity.
EQUATION
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Terminal Velocity (vₜ) occurs when:
Weight (mg) = Drag Force (Fₑ)
m = Mass of the object (kg)
g = Gravitational field strength (9.81 m s⁻²)
Fₑ = Resistive drag force (N)
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From the drag equation, terminal velocity depends on multiple parameters:
Mass (m): Heavier objects achieve higher terminal speeds because more drag is required to balance their weight.
Fluid density (ρ): Denser fluids increase drag, reducing terminal velocity.
Cross-sectional area (A): Larger areas increase drag, lowering terminal velocity.
Drag coefficient (C_d): Determined by shape and surface characteristics; streamlined shapes have smaller coefficients and higher terminal speeds.
Drag Coefficient and Shape
The drag coefficient (C_d) is a dimensionless measure of how streamlined an object is. It quantifies the efficiency of motion through a fluid.
Drag coefficient for a sphere declines with increasing Reynolds number and drops sharply at the drag crisis; a rough surface triggers the crisis earlier than a smooth one. This demonstrates how speed and surface texture alter C_d and therefore the drag force at a given area. (Extra detail beyond the syllabus: the labelled flow-regime bands on the plot.) Source
Typical values:
Sphere: ~0.47
Smooth flat plate: ~1.2
Streamlined body: ~0.04
Reducing C_d through aerodynamic or hydrodynamic design is crucial in improving speed and energy efficiency in engineering and sports applications.
Summary of Key Dependencies
Main factors influencing drag and terminal velocity:
Speed (v): Drag increases with the square of velocity.
Cross-sectional area (A): Larger area = greater drag.
Surface characteristics: Smooth surfaces reduce viscous drag; rough surfaces can reduce pressure drag at high speeds.
Fluid density (ρ): Higher density means greater resistive force.
Shape (C_d): Streamlined forms achieve smaller drag coefficients and higher terminal speeds.
Understanding how these variables interact allows physicists and engineers to predict motion through fluids, optimise performance, and ensure safe, efficient design across applications from falling bodies to aerospace vehicles.
FAQ
As air temperature increases, air density decreases because the air molecules move faster and spread out. Since drag force is proportional to air density, a lower density means less drag.
This is why objects, such as aircraft or projectiles, experience slightly less drag in warmer conditions. Conversely, in cold air, drag increases due to higher density. Although the effect is small for moderate temperature changes, it becomes more noticeable at high altitudes or in extreme weather.
The shape of an object influences how smoothly air can flow around it. Streamlined shapes minimise areas of separated flow and reduce the size of the low-pressure wake behind the object.
Non-streamlined or blunt shapes cause large turbulent wakes, increasing pressure drag. For example:
A teardrop shape minimises wake and drag.
A flat plate or cube produces high turbulence and large drag.
The smoother the flow separation, the lower the drag coefficient (C_d).
The drag crisis refers to a sudden drop in drag coefficient (C_d) observed for spheres and other rounded bodies at high Reynolds numbers.
It occurs when the boundary layer transitions from laminar to turbulent flow earlier around the object. This delays flow separation and reduces the size of the wake, lowering pressure drag dramatically.
This is the reason dimpled golf balls or roughened vehicle surfaces can experience less drag at high speeds than smooth ones.
At higher altitudes, air density decreases, reducing the resistive drag force acting on a falling object.
For the same object and weight:
Lower air density means drag builds up more slowly as speed increases.
Therefore, the object reaches a higher terminal velocity before drag equals weight.
This explains why skydivers fall faster at high altitude than near the ground, where air is denser and drag increases more rapidly.
Drag increases with the square of velocity because faster movement through a fluid forces more air (or liquid) out of the way each second, causing higher pressure and energy loss.
In contrast, solid friction depends mainly on the normal force and surface roughness, not on speed.
Thus, at high speeds, drag dominates as the principal resistive force, while at low speeds or on solid contact surfaces, friction is more significant.
Practice Questions
Question 1 (2 marks)
State two factors that affect the magnitude of the drag force experienced by an object moving through air.
Mark Scheme:
1 mark for correctly identifying speed of the object.
1 mark for correctly identifying any one of:
cross-sectional area of the object,
surface characteristics (e.g. smoothness or roughness),
shape or streamlining of the object,
density of air.
Question 2 (5 marks)
A skydiver of mass 75 kg jumps from a stationary hot air balloon. Describe and explain how the forces acting on the skydiver change during the fall until terminal velocity is reached.
Mark Scheme:
1 mark: Initially, the only significant force is weight acting downwards; the drag force is negligible because speed is very low.
1 mark: As the skydiver accelerates, speed increases, causing the drag force to increase.
1 mark: The resultant force (weight minus drag) decreases as drag increases.
1 mark: Eventually, drag becomes equal in magnitude to weight, so the net force is zero.
1 mark: At this point, the skydiver falls at constant terminal velocity because there is no further acceleration (in accordance with Newton’s First Law).
