Momentum & Impulse

Introduction to Momentum

Momentum is a fundamental concept in physics that quantifies the motion of objects with mass. It is a vector quantity, meaning it has both magnitude and direction, and is directly proportional to an object's mass and velocity.

• Definition: Momentum (p) is defined as the product of mass (m) and velocity (v).

• Formula:

• Units: kg·m/s

• Direction: Momentum always points in the same direction as the velocity of the object.

• Sign: If the object moves in the negative direction, both velocity and momentum are negative.

• Conceptual Meaning: Momentum measures how difficult it is to stop a moving object.

Example: A 4,000 kg truck moves to the right with 10 m/s. An 800 kg racecar moves to the left with 50 m/s. Their momenta are:

• Truck: kg·m/s (right)

• Racecar: kg·m/s (left)

Impulse

Impulse is the change in momentum of an object when a force is applied over a period of time. It is closely related to Newton's Second Law and is used to analyze collisions and force interactions.

• Definition: Impulse (J) is the product of force (F) and the time interval (Δt) during which the force acts.

• Formula:

• Units: N·s or kg·m/s

• Relation to Work: Both impulse and work involve force, but work is force times displacement, while impulse is force times time.

Example: You push a 50 kg crate with a constant 100 N force for 8 seconds. The impulse delivered is N·s.

Impulse from Force vs. Time Graphs

Impulse can be calculated as the area under a force vs. time graph. This is analogous to calculating work as the area under a force vs. displacement graph.

• Area Calculation: For rectangles: ; for triangles:

• Sign: Areas above the time axis represent positive impulse; areas below represent negative impulse.

Total Momentum of a System

When multiple objects interact, the total momentum of the system is the sum of the individual momenta. This is crucial for analyzing collisions and other interactions.

• Formula:

• System: A collection of objects whose total momentum is considered.

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant during interactions such as collisions or explosions.

• Principle:

• Application: Used to solve collision and push-away problems.

Types of Collisions

Collisions are classified based on whether momentum and mechanical energy are conserved, and whether objects stick together after the collision.

• Elastic Collision: Both momentum and mechanical energy are conserved. Objects do not stick together.

• Inelastic Collision: Momentum is conserved, but mechanical energy is not. Some energy is lost.

• Completely Inelastic Collision: Objects stick together and move with the same final velocity. Momentum is conserved, but mechanical energy is not.

Identifying Collision Types

To determine the type of collision, follow a series of checks based on the characteristics of each type:

• Check #1: Is momentum conserved?

• Check #2: Are final velocities equal or objects "stuck"? If yes, completely inelastic.

• Check #3: Is ? If yes, elastic; if no, inelastic.

Center of Mass

The center of mass (C.O.M) is a point representing the average position of all the mass in a system. It is useful for simplifying the analysis of systems of objects.

• Definition: The C.O.M is the weighted average of the positions of all masses in the system.

• Formula:

• Properties: The C.O.M is closer to objects with greater mass.

Example: Two objects (m1=10kg, m2=10kg) placed at x=0 and x=4.

Summary Table: Key Equations

Additional info: These notes expand on brief points and fill in missing context for clarity and completeness, suitable for exam preparation.