understand that objects moving against a resistive force may reach a terminal (constant) velocity

Momentum and Newton’s Laws of Motion

Learning Objective

Understand that an object moving against a resistive force may reach a terminal (constant) velocity.

1. Recap of Newton’s Laws

• First law (Inertia): An object remains at rest or in uniform motion unless acted on by a net external force.
• Second law: The net force on an object is proportional to the rate of change of its momentum, $$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}$$ For constant mass, this reduces to $$\mathbf{F}_{\text{net}} = m\mathbf{a}.$$
• Third law: For every action there is an equal and opposite reaction.

2. Momentum

Momentum $\mathbf{p}$ is a vector quantity defined as $$\mathbf{p}=m\mathbf{v}.$$

From Newton’s second law, the change in momentum over a time interval $\Delta t$ is related to the impulse $J$:

$$J = \int_{t_1}^{t_2}\mathbf{F}\,dt = \Delta\mathbf{p}.$$

3. Forces Acting on a Falling Object

Consider an object of mass $m$ falling vertically through a fluid (air, water, etc.). The forces are:

Force	Direction	Expression
Weight	Downwards	$\mathbf{W}=mg$
Resistive (drag) force	Upwards (opposes motion)	$\mathbf{F}_d = kv$ (linear) or $\mathbf{F}_d = \tfrac12 C\rho A v^{2}$ (quadratic)

4. Derivation of Terminal \cdot elocity

When the object reaches terminal velocity $v_t$, its acceleration becomes zero, so the net force is zero:

$$mg - F_d = 0 \quad\Longrightarrow\quad mg = F_d.$$

For a linear resistive force $F_d = kv$, solving for $v_t$ gives:

$$v_t = \frac{mg}{k}.$$

For a quadratic drag $F_d = \tfrac12 C\rho A v^{2}$, the terminal speed is:

$$v_t = \sqrt{\frac{2mg}{C\rho A}}.$$

5. Example Problem

1. A steel sphere of mass $0.15\;\text{kg}$ falls through air. The linear drag coefficient is $k = 0.025\;\text{N·s/m}$. Calculate its terminal velocity.

Solution:

$$v_t = \frac{mg}{k} = \frac{0.15 \times 9.81}{0.025} \approx 58.9\;\text{m s}^{-1}.$$

6. Key Points to Remember

• Terminal velocity occurs when the resistive force exactly balances the weight.
• At terminal speed the acceleration is zero, but the object continues to move with constant velocity.
• The form of the resistive force (linear vs quadratic) depends on the Reynolds number and the nature of the flow.

7. Suggested Diagram

8. Summary Table of Equations

Concept	Equation	When to Use
Momentum	$\mathbf{p}=m\mathbf{v}$	Any moving object
Newton’s 2nd law (general)	$\mathbf{F}_{\text{net}} = \dfrac{d\mathbf{p}}{dt}$	Variable mass or changing velocity
Newton’s 2nd law (constant mass)	$\mathbf{F}_{\text{net}} = m\mathbf{a}$	Mass does not change
Linear drag	$F_d = kv$	Low speeds, laminar flow
Quadratic drag	$F_d = \tfrac12 C\rho A v^{2}$	Higher speeds, turbulent flow
Terminal velocity (linear)	$v_t = \dfrac{mg}{k}$	When $F_d = kv$
Terminal velocity (quadratic)	$v_t = \sqrt{\dfrac{2mg}{C\rho A}}$	When $F_d = \tfrac12 C\rho A v^{2}$