define and use linear momentum as the product of mass and velocity

Momentum and Newton’s Laws of Motion

Objective

Define and use linear momentum as the product of mass and velocity.

1. Linear Momentum – Definition

The linear momentum $\mathbf{p}$ of a particle is defined as the product of its mass $m$ and its velocity $\mathbf{v}$:

$$\mathbf{p}=m\mathbf{v}$$

Momentum is a vector quantity; it has the same direction as the velocity.

2. Units and Symbols

Symbol	Quantity	SI Unit	Expression
$m$	Mass	kilogram (kg)	given
$\mathbf{v}$	Velocity	metre per second (m·s⁻¹)	given or measured
$\mathbf{p}$	Linear momentum	kilogram metre per second (kg·m·s⁻¹)	$m\mathbf{v}$

3. Connection with Newton’s Second Law

Newton’s second law can be written in terms of momentum:

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

If the mass of the object is constant, this reduces to the familiar form:

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

Thus, a net external force changes the momentum of a body.

4. Conservation of Linear Momentum

In the absence of external forces, the total linear momentum of a closed system remains constant:

$$\sum \mathbf{p}_{\text{initial}} = \sum \mathbf{p}_{\text{final}}$$

This principle is especially useful for analysing collisions and explosions.

5. Example: One‑Dimensional Elastic Collision

1. Two carts on a frictionless track: cart A (mass $m_A = 0.5\,$kg) moves at $v_A = 2.0\,$m·s⁻¹ towards cart B (mass $m_B = 0.8\,$kg) which is initially at rest.
2. Apply conservation of momentum: $$m_A v_A + m_B v_B = m_A v_A' + m_B v_B'$$ (where primed quantities are after the collision).
3. Apply conservation of kinetic energy for an elastic collision: $$\frac{1}{2}m_A v_A^2 + \frac{1}{2}m_B v_B^2 = \frac{1}{2}m_A {v_A'}^2 + \frac{1}{2}m_B {v_B'}^2$$
4. Solving the two equations gives: $$v_A' = -0.57\ \text{m·s}^{-1},\qquad v_B' = 1.43\ \text{m·s}^{-1}$$
5. Check momentum: $$0.5(2.0) + 0.8(0) = 0.5(-0.57) + 0.8(1.43) \approx 1.0\ \text{kg·m·s}^{-1}$$

6. Common Misconceptions

• Momentum is not the same as force; force is the rate of change of momentum.
• Mass must be included; “velocity alone” does not define momentum.
• Momentum is conserved only when external forces are negligible.

7. Suggested Diagram

8. Summary

Linear momentum $\mathbf{p}=m\mathbf{v}$ is a fundamental vector quantity linking mass and velocity. Newton’s second law states that the net external force equals the time rate of change of momentum. In isolated systems, total momentum is conserved, providing a powerful tool for solving collision and explosion problems.