Define resultant force as the change in momentum per unit time; recall and use the equation F = Δp / Δt

Momentum – Resultant Force

Learning Objective

Define resultant force as the change in momentum per unit time and use the equation

\$F = \frac{\Delta p}{\Delta t}\$

to solve problems.

Key Concepts

• Momentum (\$p\$) is a vector quantity defined as \$p = mv\$, where \$m\$ is mass and \$v\$ is velocity.
• The resultant (net) force acting on an object changes its momentum.
• Resultant force is defined as the rate of change of momentum:

  \$\mathbf{F}_{\text{resultant}} = \frac{\Delta \mathbf{p}}{\Delta t}\$

• If the mass is constant, the equation reduces to Newton’s second law \$F = ma\$.

Derivation of \$F = \Delta p / \Delta t\$

Starting from the definition of momentum:

\$p = mv\$

For a constant mass, the change in momentum over a time interval \$\Delta t\$ is:

\$\Delta p = m\Delta v\$

Dividing both sides by \$\Delta t\$ gives the average resultant force during that interval:

\$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = m\frac{\Delta v}{\Delta t} = ma\$

In the limit as \$\Delta t \to 0\$, the average force becomes the instantaneous force:

\$\mathbf{F} = \frac{d\mathbf{p}}{dt}\$

Units

Worked Example

Problem: A 0.15 kg ball is struck and its velocity changes from 2.0 m s⁻¹ to 8.0 m s⁻¹ in 0.05 s. Find the average resultant force exerted on the ball.

1. Calculate the initial and final momentum:

   • \$pi = m vi = 0.15 \times 2.0 = 0.30\ \text{kg·m·s}^{-1}\$
   • \$pf = m vf = 0.15 \times 8.0 = 1.20\ \text{kg·m·s}^{-1}\$

2. Find the change in momentum:

   \$\Delta p = pf - pi = 1.20 - 0.30 = 0.90\ \text{kg·m·s}^{-1}\$

3. Use the definition of resultant force:

   \$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{0.90}{0.05} = 18\ \text{N}\$

4. State the answer: The average resultant force is 18 N in the direction of the ball’s motion.

Common Mistakes to Avoid

• Confusing momentum (kg·m·s⁻¹) with kinetic energy (J). They are different physical quantities.
• Omitting the direction of the force; remember that both momentum and force are vectors.
• Using \$F = ma\$ when the mass is not constant (e.g., rockets). In such cases, use \$F = \Delta p / \Delta t\$.
• Forgetting to convert units (e.g., grams to kilograms) before substituting into the formula.

Summary

The resultant (net) force acting on an object is defined as the rate at which its momentum changes:

\$\mathbf{F}_{\text{resultant}} = \frac{\Delta \mathbf{p}}{\Delta t}\$

This relationship is fundamental for solving problems involving collisions, rockets, and any situation where mass may change or forces act over short time intervals.