Momentum and Impulse

Definitions and Principles

Momentum and impulse are fundamental concepts in physics that describe the motion of objects and how they change when forces are applied.

• Momentum (p): The product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction.

• Impulse (J): The effect of a force acting over a period of time, resulting in a change in momentum.

• Impulse-Momentum Theorem: The impulse delivered to an object is equal to the change in its momentum.

Formulas:

• Momentum:

• Impulse:

• Impulse-Momentum Theorem:

Example: When a bat strikes a ball, the force applied over the contact time changes the ball's momentum, demonstrating the impulse-momentum relationship.

Conservation Laws

General Principles

Conservation laws are foundational in physics, stating that certain quantities remain constant in isolated systems.

• Conservation of Momentum: In an isolated system (no net external force), the total momentum before an interaction equals the total momentum after.

Formula:

• Total momentum:

• Conservation:

Example: Two ice skaters push off from each other and move in opposite directions; their combined momentum remains constant if no external forces act.

Inelastic Collisions (Section 9.5)

Types and Characteristics

Collisions are classified based on how kinetic energy and momentum are conserved.

• Inelastic Collision: A collision in which kinetic energy is not conserved, but momentum is.

• Perfectly Inelastic Collision: The colliding objects stick together and move with a common final velocity.

Formula for Perfectly Inelastic Collisions:

Example: A bullet embedding itself in a block of wood is a perfectly inelastic collision.

Sample Problem: Railroad Cars

• Two railroad cars of masses kg and kg collide and stick together.

• By applying conservation of momentum, the final velocity and the initial speed of the heavier car can be determined.

Calculation Steps:

• Set up the conservation equation:

• Solve for the unknown initial velocity if other values are given.

Momentum and Collisions in Two Dimensions (Section 9.6)

Principles and Problem-Solving

When collisions occur in two dimensions, momentum must be conserved in each direction independently.

• Component Conservation: and

• Each object’s momentum is broken into x and y components.

Example: A falcon swoops down at an angle and catches a pigeon flying horizontally. The combined velocity after the collision is found by conserving momentum in both x and y directions.

Calculation Steps:

• Find initial momentum components for each object.

• Set up conservation equations for x and y directions.

• Solve for the final velocity magnitude and direction using the Pythagorean theorem and trigonometry.

Formulas:

• Speed:

• Angle:

Angular Momentum (Section 9.7)

Definitions and Conservation

Angular momentum is the rotational analog of linear momentum, important for systems involving rotation.

• Angular Momentum (L): For a rotating object, , where is the moment of inertia and is the angular velocity.

• SI Units: kg·m2/s

• Conservation of Angular Momentum: If no net external torque acts on a system, its total angular momentum remains constant.

Formulas:

• Angular momentum:

• Conservation:

Example: A figure skater spins faster by pulling in their arms, reducing their moment of inertia and increasing angular velocity to conserve angular momentum.

Changing Moment of Inertia

• Moment of inertia depends on mass distribution relative to the axis of rotation.

• When the distribution changes (e.g., a person moves outward on a merry-go-round), the moment of inertia changes, affecting angular velocity.

Sample Problem: Merry-Go-Round

• When a person moves from the center to the edge, the system's moment of inertia increases, so the angular velocity decreases to conserve angular momentum.

• Period increases as .

Comparison Table: Rotational and Linear Dynamics

Summary: Problem-Solving Strategies

• Choose an isolated system or one isolated during the interaction.

• Draw diagrams showing the system before and after the event.

• Write conservation equations for momentum (linear or angular) in vector components.

• Check if the result is reasonable (e.g., direction, magnitude).

Key Concepts and Applications

• Impulse: Area under a force vs. time curve; changes momentum.

• Collisions: Inelastic (objects stick together), elastic (objects bounce apart), and two-dimensional cases.

• Angular Momentum: Conserved in absence of external torque; changes with moment of inertia.

Applications: Vehicle collisions, animal motion (e.g., falcons and prey), rotating systems (e.g., merry-go-rounds, skaters).