understand that mass is the property of an object that resists change in motion

Momentum and Newton’s Laws of Motion

Learning Objective

Understand that mass is the property of an object that resists change in motion (inertia).

Key Concepts

• Force, mass and acceleration are linked by Newton’s second law.
• Momentum ($\mathbf{p}=m\mathbf{v}$) combines mass and velocity into a single vector quantity.
• Conservation of momentum follows from Newton’s third law.
• Mass quantifies an object’s inertia – its resistance to a change in state of motion.

Newton’s First Law – Law of Inertia

An object remains at rest or moves with constant velocity unless acted upon by a net external force. The tendency to maintain its state of motion is called inertia, and the magnitude of inertia is measured by the object's mass.

Newton’s Second Law – Relationship Between Force, Mass and Acceleration

The net force $\mathbf{F}_{\text{net}}$ on an object is proportional to the rate of change of its momentum:

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

For a body of constant mass this reduces to the familiar form:

$$\mathbf{F}_{\text{net}} = m\mathbf{a}$$

Here, $m$ is the inertial mass – the larger the mass, the smaller the acceleration produced by a given force.

Newton’s Third Law – Action and Reaction

For every action force there is an equal and opposite reaction force:

$$\mathbf{F}_{AB} = -\mathbf{F}_{BA}$$

This principle underlies the conservation of momentum in isolated systems.

Momentum

Momentum is defined as the product of an object’s mass and its velocity:

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

Because mass appears directly in the definition, a larger mass means a larger momentum for the same speed, and therefore a greater resistance to changes in motion.

Conservation of Momentum

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

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

This principle can be derived from Newton’s third law and is a powerful tool for analysing collisions.

Mass as a Measure of Inertia

Inertia is the qualitative description of an object’s resistance to a change in its state of motion. Quantitatively, inertia is expressed by the object's mass $m$. The larger the mass, the greater the force required to achieve a given acceleration, as shown by $F = ma$.

Illustrative Example

1. A 2.0 kg cart is pushed with a constant horizontal force of 10 N. What is its acceleration?
2. A 0.5 kg ball moving at 4 m s⁻¹ collides elastically with a 1.5 kg ball initially at rest. Determine the velocities after the collision using conservation of momentum and kinetic energy.

These problems highlight how mass governs the response of objects to applied forces and how momentum is conserved in interactions.

Summary Table

Law	Mathematical Form	Physical Meaning	Role of Mass
Newton’s First Law	$\mathbf{F}_{\text{net}} = 0 \;\Rightarrow\; \mathbf{v}= \text{constant}$	Objects maintain their state of motion unless acted upon.	Mass determines the amount of inertia resisting a change.
Newton’s Second Law	$\mathbf{F}_{\text{net}} = m\mathbf{a}$	Force produces acceleration proportional to mass.	Mass is the proportionality constant linking force and acceleration.
Newton’s Third Law	$\mathbf{F}_{AB} = -\mathbf{F}_{BA}$	Forces occur in equal and opposite pairs.	Mass of each body determines the resulting accelerations from the pair of forces.
Momentum Definition	$\mathbf{p}=m\mathbf{v}$	Momentum combines mass and velocity into a conserved quantity.	Mass directly scales momentum, increasing resistance to change.
Conservation of Momentum	$\sum \mathbf{p}_{\text{initial}} = \sum \mathbf{p}_{\text{final}}$	In an isolated system, total momentum remains constant.	Mass distribution among objects determines how momentum is shared after interactions.

Key Take‑aways

• Mass is the quantitative measure of an object’s inertia.
• Higher mass → greater resistance to changes in speed or direction.
• Newton’s second law links force, mass and acceleration, while momentum links mass and velocity.
• Conservation of momentum provides a powerful method for solving collision problems.