Momentum

Definition and Properties 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, and is directly proportional to both the mass and velocity of an object.

• Definition: Momentum is the product of an object's mass and its velocity.

• Symbol: p

• Equation:

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

• Vector Nature: The direction of momentum is the same as the direction of velocity.

Example: A 2 kg object moving at 3 m/s has a momentum of .

Comparing Momentum

• A moving boulder has more momentum than a stone rolling at the same speed due to its greater mass.

• A fast boulder has more momentum than a slow boulder of the same mass because of its higher velocity.

• Large ships, despite moving slowly, can have more momentum than a fast-moving bullet due to their enormous mass.

Example: A cargo ship (large mass, low speed) can have more momentum than a bullet (small mass, high speed).

Properties of Moving Objects

• Every moving object has:

  • Momentum

  • Kinetic energy

  • Speed

• All of the above properties are present in any object in motion.

Impulse

Definition and Formula

Impulse is the product of the force applied to an object and the time interval over which the force is applied. It quantifies the effect of a force acting over time and is responsible for changing an object's momentum.

• Symbol: J

• Equation:

• Unit: Newton-second (N·s), which is equivalent to kg·m/s

Example: If a force of 10 N acts on an object for 2 seconds, the impulse is .

Impulse-Momentum Theorem

Relationship Between Impulse and Momentum

The impulse-momentum theorem states that the impulse on an object is equal to the change in its momentum.

• Equation:

• Or, equivalently:

• Impulse and change in momentum are both vector quantities and point in the same direction as the applied force.

Example: If a 1 kg ball increases its velocity from 2 m/s to 5 m/s, the change in momentum is , so the impulse is also 3 N·s.

Impulse in Practice

• Increasing the time over which a force acts reduces the force required to achieve the same change in momentum.

• Examples include:

  • Bending your knees when landing from a jump increases the time interval, reducing the force on your legs.

  • Extending your hand when catching a ball increases the time over which the ball's momentum is brought to zero, reducing the force felt.

  • Car airbags increase the time over which the passenger's momentum is reduced to zero, decreasing the force experienced.

Example: A dish dropped on a carpet experiences a smaller force than one dropped on a hard floor because the carpet increases the stopping time, reducing the force.

Conservation of Momentum

Law of Conservation of Momentum

The law of conservation of momentum states that if no net external force acts on a system, the total momentum of the system remains constant.

• Equation:

• Applies to all types of collisions and interactions, provided external forces are negligible.

• Momentum is conserved in both magnitude and direction (vector quantity).

Example: Before and after a gun is fired, the total momentum of the gun and bullet system remains the same (the gun recoils backward as the bullet moves forward).

Collisions

Types of Collisions

• Elastic Collision: Colliding objects rebound without lasting deformation or generation of heat. Both momentum and kinetic energy are conserved.

• Inelastic Collision: Colliding objects become deformed and/or generate heat. Momentum is conserved, but kinetic energy is not.

• Perfectly Inelastic Collision: Colliding objects stick together after the collision, moving as a single combined mass.

Examples and Applications

• Two freight cars of equal mass: If one is moving and collides with another at rest, and they couple together, their combined speed after collision is half the initial speed of the moving car (assuming equal mass and a perfectly inelastic collision).

• Fish Example: A 5 kg fish swimming at 1 m/s swallows a smaller fish at rest. The velocity after the collision can be found using conservation of momentum: If the smaller fish is not at rest, its velocity is included in the calculation.

Summary Table: Types of Collisions

Additional info: The notes reference textbook problems (e.g., 24, 25, 26, 28, 31, 41 P104-106) for further practice, which likely involve calculations using the above principles.