Describe the motion of objects falling in a uniform gravitational field with and without air/liquid resistance, including reference to terminal velocity

1.2 Motion – Falling Objects

Learning Objective

Describe the motion of objects falling in a uniform gravitational field with and without air or liquid resistance, including reference to terminal velocity.

Key Concepts

• Uniform gravitational field: acceleration due to gravity \$g \approx 9.81\ \text{m s}^{-2}\$ near the Earth's surface.
• Free fall: motion under gravity alone (no resistance).
• Resisted fall: motion under gravity plus a resistive force that depends on speed and the medium.
• Terminal velocity: the constant speed reached when the net force on a falling object becomes zero.

Free Fall (No Resistance)

When air or liquid resistance is negligible, the only force acting is the weight \$W = mg\$, giving a constant acceleration \$a = g\$ downwards.

Equations of motion (starting from rest, \$u = 0\$):

\$v = gt\$

\$s = \frac{1}{2}gt^{2}\$

where \$v\$ is the instantaneous speed, \$t\$ is the time elapsed, and \$s\$ is the distance fallen.

Resisted Fall (With Air or Liquid Resistance)

In a real medium a resistive force \$F_{r}\$ opposes the motion. For many practical situations the resistance can be approximated by:

• Linear drag: \$F_{r}=kv\$ (valid for low speeds, small objects in viscous fluids).
• Quadratic drag: \$F_{r}=kv^{2}\$ (valid for higher speeds, objects moving through air).

The net force is then:

\$F{\text{net}} = mg - F{r}\$

and the resulting acceleration is:

\$a = g - \frac{F_{r}}{m}\$

Terminal \cdot elocity

Terminal velocity \$v{t}\$ occurs when \$F{\text{net}} = 0\$, i.e. when the resistive force exactly balances the weight.

For linear drag:

\$mg = kv{t} \quad\Rightarrow\quad v{t}= \frac{mg}{k}\$

For quadratic drag:

\$mg = kv{t}^{2} \quad\Rightarrow\quad v{t}= \sqrt{\frac{mg}{k}}\$

At \$v = v_{t}\$ the acceleration becomes zero and the object continues to fall at a constant speed.

Comparison of Free Fall and Resisted Fall

Factors Influencing Terminal \cdot elocity

• Mass \$m\$: heavier objects have larger \$v_{t}\$ (directly proportional for linear drag, proportional to \$\sqrt{m}\$ for quadratic drag).
• Cross‑sectional area \$A\$: larger area increases drag coefficient \$k\$, reducing \$v_{t}\$.
• Shape and surface texture: affect the drag coefficient.
• Density and viscosity of the medium: more viscous fluids increase resistance.

Example Calculation (Quadratic Drag)

Given: a steel sphere of mass \$0.5\ \text{kg}\$ falling through air with drag constant \$k = 0.25\ \text{kg m}^{-1}\$.

Find the terminal velocity.

\$v_{t}= \sqrt{\frac{mg}{k}} = \sqrt{\frac{0.5 \times 9.81}{0.25}} = \sqrt{19.62}= 4.43\ \text{m s}^{-1}\$

The sphere will accelerate until it reaches about \$4.4\ \text{m s}^{-1}\$, after which it will continue to fall at this constant speed.

Common Misconceptions

• “All objects fall at the same speed.” – True only in vacuum; in air, resistance causes different terminal velocities.
• “Terminal velocity means the object stops falling.” – Incorrect; it means the speed becomes constant, not zero.
• “Heavier objects always fall faster.” – In a resisting medium, heavier objects have higher \$v_{t}\$, but the initial acceleration is still \$g\$ for all objects.

Suggested Diagram

Summary

1. In a uniform gravitational field, an object in free fall accelerates at \$g\$ and its speed increases linearly with time.
2. When resistance is present, the resistive force grows with speed, reducing the net acceleration.
3. Terminal velocity is reached when \$mg = F_{r}\$; the object then falls at a constant speed.
4. Terminal velocity depends on mass, shape, area, and the properties of the surrounding medium.