Law of Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act upon it

Table of Contents:

• Law of Conservation of Momentum
• Derivation
• Examples of Law of Conservation of Momentum
• FAQs

Law of Conservation of Momentum

The total momentum of all objects in the system before an interaction is equal to the total momentum of all objects in the system after the interaction, provided that there are no external forces acting on the system.

Mathematically, this can be expressed as:

Σp_i = Σp_f

where Σp_i represents the initial total momentum of the system and Σp_f represents the final total momentum of the system.

This law is an important concept in physics, as it helps us understand the behavior of objects in motion, such as collisions or explosions. By using the law of conservation of momentum, we can make predictions about the motion of objects before and after an interaction, without needing to know the exact details of the interaction itself.

Derivation

The law of conservation of momentum can be derived from Newton’s laws of motion. Specifically, it is based on Newton’s third law, which states that for every action, there is an equal and opposite reaction.

Consider a system of two objects, A and B, with masses m1 and m2, respectively. Let v1i and v2i be their initial velocities before they interact. Suppose the objects collide with each other, and as a result, their velocities change to v1f and v2f, respectively.

According to Newton’s second law, the force acting on an object is equal to its mass times its acceleration. In this case, the forces acting on objects A and B during the collision are equal in magnitude and opposite in direction, as dictated by Newton’s third law. Therefore:

F_A = -F_B

where F_A and F_B are the forces acting on objects A and B, respectively.

By Newton’s second law, we have:

F_A = m1 * (v1f – v1i) / t

F_B = m2 * (v2f – v2i) / t

where t is the time for which the force acts.

If we multiply each equation by t and rearrange, we get:

m1 * (v1f – v1i) = -m2 * (v2f – v2i)

Dividing both sides by the time t, we obtain:

m1 * (v1f – v1i) / t = -m2 * (v2f – v2i) / t

Recall that the change in momentum of an object is given by:

Δp = m * (v_f – v_i)

where Δp is the change in momentum, m is the mass of the object, v_f is the final velocity, and v_i is the initial velocity.

Substituting this expression into our previous equation, we obtain:

Δp1 = -Δp2

This equation shows that the change in momentum of object A is equal in magnitude but opposite in direction to the change in momentum of object B. In other words, the total momentum of the system is conserved. This is the law of conservation of momentum.

Note that this derivation assumes that there are no external forces acting on the system during the collision. If there are external forces, then the momentum of the system may not be conserved.

Examples of Law of Conservation of Momentum

The law of conservation of momentum has many real-world applications. Here are some examples:

Elastic collisions: When two billiard balls collide on a pool table, their total momentum is conserved. This is because there are no external forces acting on the system, and the collision is elastic, meaning that the kinetic energy is also conserved.

Rockets: The law of conservation of momentum is used in rocket propulsion. As fuel is expelled out of the rocket’s engines, it generates a force that propels the rocket forward. This force is equal and opposite to the force acting on the fuel, and by Newton’s third law, it results in a change in the rocket’s momentum.

Airbags: The law of conservation of momentum is also used in the design of car airbags. When a car collides with an object, the passengers’ momentum changes rapidly. The airbag inflates and absorbs some of the force of the impact, which reduces the passengers’ momentum and helps to prevent injury.

Planetary motion: The law of conservation of momentum explains the orbits of planets around the sun. The gravitational force between the planets and the sun results in a change in momentum, but the total momentum of the system is conserved.

Atomic physics: The law of conservation of momentum is important in the study of atomic and subatomic particles. When particles collide, their momentum changes in accordance with the law of conservation of momentum. This helps scientists to understand the behavior of these particles and how they interact with each other.

Law of Conservation of Momentum FAQS