Momentum

1.6 Conservation of Momentum in One Dimension

Objective: Apply the principle of the conservation of momentum to solve simple problems in one dimension.

Understanding Momentum

Momentum is a measure of the mass of a moving object. It is defined as the product of the mass and the velocity of the object.

Equation:

$p = mv$

where:

• $p$ = momentum (kg m/s)
• $m$ = mass (kg)
• $v$ = velocity (m/s)

Newton's Third Law and Momentum

Newton's Third Law states that for every action, there is an equal and opposite reaction. This law is fundamental to understanding the conservation of momentum.

When two objects interact, the total momentum of the system remains constant, provided no external forces act on the system.

Conservation of Momentum Principle

The principle of conservation of momentum states that in a closed system (where no external forces act), the total momentum before an event is equal to the total momentum after the event.

Equation:

$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

where:

• $m_1$ and $m_2$ are the masses of the two objects.
• $v_1$ and $v_2$ are the initial velocities of the two objects.
• $v_1'$ and $v_2'$ are the final velocities of the two objects.

Examples and Problem Solving

Let's consider some examples to illustrate how to apply the conservation of momentum principle.

Example 1: Collision of Two Objects

Two objects with masses $m_1$ and $m_2$ collide. Initially, object 1 has a velocity $v_1$ and object 2 has a velocity $v_2$. After the collision, object 1 has a velocity $v_1'$ and object 2 has a velocity $v_2'$.

Applying the conservation of momentum principle:

$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

Example 2: Explosion

An object with mass $m$ is at rest. It explodes into two pieces with masses $m_1$ and $m_2$ and velocities $v_1$ and $v_2$ respectively.

Applying the conservation of momentum principle:

$0 = m_1v_1 + m_2v_2$

Table Summary

Concept	Equation	Description
Momentum	$p = mv$	Measure of the motion of an object.
Newton's Third Law	Action = Reaction	For every action, there is an equal and opposite reaction.
Conservation of Momentum	$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$	Total momentum of a closed system remains constant.

By understanding the principle of conservation of momentum, you can analyze and solve various problems involving collisions and explosions in one dimension.