Unit 3 · Topic 06

Fluid resistance

A body moving relative to a fluid can experience a drag force opposite that relative velocity. Its dependence on speed changes with flow regime and geometry.

Unit 3DragOpen navigation

Concept 01

Drag force

Essential The minimum you should retain

Fluid resistance acts against relative motion between a body and the fluid. If the air itself moves, the relevant velocity is the body's velocity relative to the air, not necessarily relative to the ground.

UnderstandInterpret and connect

At low speeds in some viscous regimes, a linear model F_D = -b v_rel is useful. In other regimes, a force approximately quadratic in speed is common.

DeepenFormulation and conditions

A useful vector form of quadratic drag is F_D = -c |v_rel| v_rel. The magnitude is c v_rel² and the vector form guarantees opposition to the relative velocity.

ExploreConnections for further study

Coefficients b and c depend on fluid, shape, size, and flow regime. They are not universal constants. A change of regime can change the functional form itself.

Mathematical relation

Linear drag model

F→D=−bv→rel \vec F_D=-b\vec v_{rel}
Represents

A linear drag force opposite velocity relative to the fluid.

Physical interpretation

The minus sign sets the direction opposite relative velocity.

DeepenVariables, conditions, and checks

Variables

F⃗_D
drag force; usual unit: N
b
linear coefficient; usual unit: kg/s
v⃗_rel
velocity relative to the fluid; usual unit: m/s

Conditions of application

  • A regime where the linear approximation is valid has been declared.

Dimensional check

(kg/s)(m/s)=N.

Errors it helps prevent

  • Using ground velocity.
  • Treating one drag law as universal.

Mathematical relation

Quadratic drag model

F→D=−c|v→rel|v→rel \vec F_D=-c|\vec v_{rel}|\vec v_{rel}
Represents

A force of magnitude cv_rel² opposite relative velocity.

Physical interpretation

The |v_rel| factor gives quadratic dependence while retaining vector direction.

DeepenVariables, conditions, and checks

Variables

c
quadratic coefficient; usual unit: kg/m
v⃗_rel
velocity relative to the fluid; usual unit: m/s

Conditions of application

  • A regime where the quadratic approximation is valid has been declared.

Dimensional check

(kg/m)(m²/s²)=N.

Errors it helps prevent

  • Using ground velocity.
  • Treating one drag law as universal.

Drag magnitude versus speed

A line and parabola show two possible dependencies of drag magnitude on speed.
  • bv, b=2 kg/s
  • cv², c=0.25 kg/m

With illustrative coefficients, bv grows linearly and cv² quadratically. These are regime models, not universal experimental data.

Concept 02

Terminal speed

Essential The minimum you should retain

A falling body in a fluid may accelerate at first and then approach a constant speed when drag balances weight.

UnderstandInterpret and connect

Terminal speed occurs when net force is zero, not when gravity disappears. Weight still acts and is balanced by drag and, in a more complete model, by other forces such as buoyancy.

DeepenFormulation and conditions

For a simple vertical model neglecting buoyancy, linear drag gives terminal speed magnitude v_t = mg/b; quadratic drag gives v_t = sqrt(mg/c). These results depend on the stated model.

ExploreConnections for further study

The approach to v_t can be gradual. With linear drag and downward positive, m dv/dt = mg - bv. Its solution contains a time scale m/b, although deriving the full solution is not a core requirement here.

Worked example

Terminal speed with quadratic drag

A 0.20 kg body falls with quadratic drag magnitude cv², where c = 0.050 kg/m. Buoyancy is ignored.

Given
  • m = 0.20 kg
  • c = 0.050 kg/m
  • g = 9.8 m/s²
Target

Find the terminal-speed magnitude.

  1. Terminal state

    At terminal speed acceleration is zero.

  2. Balance

    mg - cv_t² = 0.

  3. Solve

    v_t = sqrt(mg/c).

  4. Calculation

    v_t = sqrt(0.20·9.8/0.050) = sqrt(39.2) ≈ 6.26 m/s.

Conclusion

Terminal speed has magnitude about 6.3 m/s downward.

Mathematical relation

Terminal speed with linear drag

vt=mgb v_t=\frac{mg}{b}
Represents

Terminal-speed magnitude under linear drag.

Physical interpretation

Gravity still acts and drag balances it.

DeepenVariables, conditions, and checks

Variables

v_t
terminal speed; usual unit: m/s
b
linear coefficient; usual unit: kg/s

Conditions of application

  • Vertical fall.
  • Linear drag.
  • Buoyancy neglected.
  • Terminal state with a=0.

Dimensional check

(kg·m/s²)/(kg/s)=m/s.

Errors it helps prevent

  • Claiming gravity disappears.

Mathematical relation

Terminal speed with quadratic drag

vt=mgc v_t=\sqrt{\frac{mg}{c}}
Represents

Terminal-speed magnitude under quadratic drag.

Physical interpretation

The square root follows from mg=cv_t².

DeepenVariables, conditions, and checks

Variables

v_t
terminal speed; usual unit: m/s
c
quadratic coefficient; usual unit: kg/m

Conditions of application

  • Vertical fall.
  • Quadratic drag.
  • Buoyancy neglected.
  • Terminal state with a=0.

Dimensional check

The radicand has dimension m²/s².

Errors it helps prevent

  • Claiming gravity disappears.
Three fall states show the same weight and increasing drag arrows until drag equals weight at terminal speed.low speedintermediateterminalmgF_DmgF_DmgF_D

As speed increases, drag grows: first it is smaller than weight and at terminal state it matches weight. Gravity never disappears.

Concept 03

Model versus reality

Essential The minimum you should retain

There is no single drag formula valid for every fluid and every speed.

UnderstandInterpret and connect

Before using b v or c v², state the model. A numerical answer without that assumption may look precise while representing the system poorly.

DeepenFormulation and conditions

Flow regime is often characterized by dimensionless parameters such as Reynolds number. This unit uses that idea only to explain why drag laws change; it does not develop fluid dynamics.

ExploreConnections for further study

Comparing a model with experimental data can estimate drag coefficients and identify the speed range over which an approximation is useful.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using ground speed when the fluid itself is moving.

Use the body's velocity relative to the fluid.

Applying bv or cv² without stating the regime.

A drag law is a regime-dependent approximation.

Claiming gravity no longer acts at terminal speed.

Gravity still acts; net force vanishes because forces balance.