Momentum and Force

Definition of Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Momentum (\(\vec{p}\)) is defined as the product of an object's mass and its velocity:

• Units: kilogram meter per second (kg·m/s)

Newton's Second Law in Terms of Momentum

Newton originally formulated his second law of motion in terms of momentum. The law states:

• The rate of change of momentum of an object is equal to the net force applied to it.

• This equation shows that force is related to the time rate of change of momentum, just as acceleration is the time rate of change of velocity.

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

• Key Point: Momentum provides a more general framework for analyzing motion, especially when mass is not constant (e.g., rockets).

Collisions

Types of Collisions

Collisions are classified based on whether kinetic energy is conserved during the event.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. Some energy is transformed into other forms (e.g., heat, sound).

• Completely Inelastic Collision: The colliding objects stick together after the collision, moving as a single mass.

Elastic Collisions

In an elastic collision, the total kinetic energy and momentum of the system are conserved.

• Kinetic Energy Conservation:

• Velocity Relation (for 1D elastic collisions):

• Momentum Conservation: Must also be satisfied for all collisions.

Inelastic Collisions

In inelastic collisions, kinetic energy is not conserved. The lost kinetic energy is converted into other forms, such as heat or deformation.

• Energy Conservation (including other forms):

• Momentum Conservation:

• Completely Inelastic Collision: The objects stick together, and their combined mass moves with a common velocity after the collision.

Collisions in Two Dimensions

Conservation of Momentum in Multiple Dimensions

When collisions occur in two (or more) dimensions, momentum conservation must be applied separately to each component (x and y directions).

• Momentum is a vector: It can be broken into x- and y-components.

• Conservation Equations:

• Each component must be conserved independently.

Example: Billiard Balls Collision

Suppose ball A (mass m) moves at 3.0 m/s in the +x direction and strikes ball B (also mass m) at rest. After the collision, both balls move off at 45° above and below the x-axis, respectively.

• Conservation in x-direction:

• Conservation in y-direction:

• By solving these equations, the final velocities can be determined.

Center of Mass

Definition and Calculation

The center of mass (CM) of a system is the point at which the mass of the system can be considered to be concentrated for the purposes of analyzing translational motion.

• For two particles:

• For many particles:

• For a continuous object:

• Where is the total mass and is an infinitesimal mass element at position .

Center of Mass of a Uniform Rod

For a uniform rod of length and mass , the center of mass is at its midpoint:

• For non-uniform density: If the density varies along the length, integrate using the density function .

Motion of the Center of Mass

The motion of the center of mass for a system of particles is determined by the net external force acting on the system.

• Velocity of the center of mass:

• Acceleration of the center of mass:

• Newton's Second Law for a system: The net external force equals the total mass times the acceleration of the center of mass.

Summary Table: Types of Collisions

Key Equations

• Momentum:

• Newton's Second Law (momentum form):

• Kinetic Energy:

• Conservation of Momentum (1D):

• Center of Mass (discrete):

• Center of Mass (continuous):

Example: Ballistic Pendulum

A ballistic pendulum is used to measure the speed of a projectile. A projectile of mass is fired into a block of mass suspended as a pendulum. After the collision, the combined mass swings upward to a height .

• Step 1: Use conservation of momentum to find the velocity just after the collision:

• Step 2: Use conservation of energy to relate the velocity to the height:

• Step 3: Solve for the initial velocity of the projectile.

Summary of Key Points

• Momentum is a vector quantity defined as .

• Newton's second law relates force to the rate of change of momentum.

• Momentum is always conserved in isolated systems; kinetic energy is only conserved in elastic collisions.

• In completely inelastic collisions, objects stick together after the collision.

• The center of mass is the point where the total mass of a system can be considered to act for translational motion.