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{t1}^{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:

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

For a linear resistive force \$Fd = kv\$, solving for \$vt\$ 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