Terminal Velocity and Drag

Drag (or air resistance) is a force that opposes the motion of an object through a fluid (gas or liquid).

Key properties of drag:

• Acts in the opposite direction to motion
• Is a contact force (the object touches the fluid particles)
• Increases as the speed of the object increases
• Increases with greater surface area (cross-sectional area)
• Depends on the shape of the object (streamlined shapes reduce drag)
• Depends on the density of the fluid (water creates more drag than air)

Consider a skydiver jumping from a plane. Let's track the forces at each stage:

Stage 1: Just After Jumping

• Weight acts downwards (due to gravity)
• Air resistance is very small (speed is low)
• Resultant force is large, downwards
• The skydiver accelerates downwards rapidly

Stage 2: Picking Up Speed

• Weight stays the same (mass and $g$ don't change)
• Speed increases, so air resistance increases
• Resultant force is still downwards, but smaller than before
• The skydiver is still accelerating, but the acceleration is decreasing

Stage 3: Terminal Velocity Reached

• Air resistance has grown until it equals the weight
• Resultant force = 0 (forces are balanced)
• Acceleration = 0
• The skydiver moves at constant velocity — this is terminal velocity

$\text{At terminal velocity: } W = F_{\text{drag}}$

Stage 4: Parachute Opens

• The parachute dramatically increases the surface area
• Air resistance suddenly increases massively
• Air resistance is now much greater than weight
• Resultant force is upwards (opposing the downward motion)
• The skydiver decelerates rapidly

Stage 5: New Terminal Velocity

• As speed decreases, air resistance decreases
• Eventually air resistance equals weight again
• A new, much lower terminal velocity is reached
• The skydiver lands safely at about 5–6 m/s

The motion described above produces a characteristic v-t graph:

1. Steep curve upward — rapid acceleration at the start
2. Curve flattening — acceleration decreasing as drag increases
3. Horizontal line — terminal velocity reached
4. Sharp drop — parachute opens, rapid deceleration
5. Short steep section, then new horizontal line — new (lower) terminal velocity

Examples:

• A feather has low mass but high surface area → very low terminal velocity
• A skydiver with an open parachute has much higher surface area → much lower terminal velocity
• A steel ball has high mass and small surface area → very high terminal velocity

Terminal velocity also occurs in liquids. A ball bearing dropped into a tube of viscous liquid (like glycerol or oil):

1. Initially accelerates (weight > drag + upthrust)
2. As speed increases, drag increases
3. Eventually reaches terminal velocity when: weight = drag + upthrust

This is the basis of Stokes' Law (A-Level topic).

Without air resistance, ALL objects fall at the same rate regardless of mass. This was demonstrated by Galileo (and confirmed by astronaut David Scott on the Moon with a hammer and feather).

In a vacuum, there is no air resistance, so:

• All objects accelerate at $g = 9.8$ m/s²
• There is no terminal velocity
• A feather and a bowling ball hit the ground at the same time

In air, different objects have different terminal velocities because of different masses and surface areas.

• Drag/air resistance opposes motion through a fluid; increases with speed and surface area
• Terminal velocity is reached when drag = weight (resultant force = 0)
• Object accelerates at first, then acceleration decreases, then constant velocity
• Opening a parachute increases surface area → higher drag → lower terminal velocity
• In a vacuum, there is no drag, so no terminal velocity
• On v-t graphs: curve flattening to a horizontal line shows terminal velocity