describe and explain qualitatively the motion of objects in a uniform gravitational field with air resistance

Momentum and Newton’s Laws of Motion

Learning Objective

Describe and explain qualitatively the motion of objects in a uniform gravitational field when air resistance is present.

Key Concepts

• Momentum, \$p = mv\$
• Newton’s First Law – inertia
• Newton’s Second Law – \$F = ma = \dfrac{dp}{dt}\$
• Newton’s Third Law – action–reaction pairs
• Uniform gravitational field, \$g = 9.81\ \text{m s}^{-2}\$
• Air‑resistance (drag) forces

Newton’s Second Law in \cdot ector Form

For a particle of mass \$m\$ moving with velocity \$\mathbf{v}\$, the net external force \$\mathbf{F}_{\text{net}}\$ satisfies

\$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}= m\frac{d\mathbf{v}}{dt}= m\mathbf{a}\$

Forces on a Falling Object

When an object moves vertically downwards in air, two main forces act:

1. Weight: \$\mathbf{W}=m\mathbf{g}\$ (directed downwards)
2. Air‑resistance (drag): \$\mathbf{F}_{\text{drag}}\$ (directed upwards, opposite to motion)

Typical Forms of Drag

• Linear drag (low speeds, small objects): \$F_{\text{drag}} = k v\$
• Quadratic drag (higher speeds, larger objects): \$F{\text{drag}} = \tfrac{1}{2} Cd \rho A v^{2}\$

Qualitative Motion with Air Resistance

Consider the vertical motion of a body released from rest. The net force at any instant is

\$F{\text{net}} = mg - F{\text{drag}}(v)\$

Three distinct regimes can be identified:

Terminal \cdot elocity

When \$mg = F{\text{drag}}(vt)\$, the object no longer accelerates. Solving for \$v_t\$ gives:

• Linear drag: \$v_t = \dfrac{mg}{k}\$
• Quadratic drag: \$vt = \sqrt{\dfrac{2mg}{Cd \rho A}}\$

Momentum Perspective

Using \$F_{\text{net}} = dp/dt\$, the rate of change of momentum is directly linked to the imbalance between weight and drag. As the object approaches terminal velocity, \$dp/dt \rightarrow 0\$, indicating a steady momentum state.

Factors Influencing the Motion

• Mass \$m\$: Heavier objects have larger \$v_t\$ because weight grows faster than drag.
• Cross‑sectional area \$A\$: Larger \$A\$ increases drag, reducing \$v_t\$.
• Shape (drag coefficient \$Cd\$): Streamlined shapes have lower \$Cd\$, giving higher \$v_t\$.
• Air density \$\rho\$: Higher \$\rho\$ (e.g., at lower altitude) increases drag.

Example Scenario (Qualitative)

A feather and a steel ball are dropped from the same height in still air.

1. Initially, both experience \$mg\$ downwards, but the feather’s tiny mass means \$mg\$ is comparable to its drag almost immediately.
2. The steel ball, being much heavier, retains a net downward force for a longer time, accelerating close to \$g\$ before drag becomes significant.
3. Both eventually reach terminal velocities, but \$v_t\$ for the steel ball is far larger than for the feather, so the ball reaches the ground much sooner.

Common Misconceptions

• “Air resistance always reduces acceleration to zero.” – It only does so when the object reaches terminal velocity; before that, acceleration is reduced but not eliminated.
• “Heavier objects fall faster because of greater weight.” – In vacuum, yes; in air, the larger weight also increases the drag force needed to balance it, giving a higher terminal speed.
• “Momentum is conserved in free fall.” – Momentum changes because external forces (gravity and drag) act on the object.

Suggested Diagram

Summary

• Newton’s second law links net force to the change in momentum.
• In a uniform gravitational field, weight provides a constant downward force.
• Air resistance opposes motion; its magnitude grows with speed (linear or quadratic).
• The competition between \$mg\$ and \$F_{\text{drag}}\$ creates three motion regimes, ending in a constant‑speed terminal velocity where \$dp/dt = 0\$.
• Mass, area, shape, and air density determine the quantitative value of terminal velocity.