Momentum and Collisions

Definition and Conservation of Momentum

Momentum is a fundamental concept in physics, describing the quantity of motion an object possesses. It is conserved in isolated systems, especially during collisions.

• Momentum (p): Defined as mass times velocity:

• Net Force and Momentum: A net force acting on a system changes its momentum according to Newton's Second Law.

• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.

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

• Inelastic Collisions: Momentum is conserved, but kinetic energy is not necessarily conserved.

Example: Elastic Collision

Consider two curling stones colliding on ice:

• Let stone 1 (mass ) move at and stone 2 (mass ) at .

• Conservation of momentum:

• Conservation of kinetic energy:

• If masses are equal and the collision is elastic, the angle between the two final velocities is 90 degrees.

Momentum and Kinetic Energy Relationship

• Kinetic energy:

• Momentum squared:

• Relationship:

• For speeds greater than 0.1c (where c is the speed of light), use special relativity:

Center of Mass

Definition and Importance

The center of mass of a system of particles is the point where the system's mass can be considered to be concentrated for the analysis of translational motion.

• Helps simplify problems involving many particles.

• Motion can be separated into motion of the center of mass and motion about the center of mass.

Center of Mass Equations

• For two particles in 1D:

• For 2D and 3D systems:

Examples

• Hydrogen Atom: Proton at , electron at m. is very close to the proton due to its much larger mass.

• Earth-Moon System: The center of mass is inside the Earth, closer to the larger mass.

Application: Reduced Mass

In two-body systems, the reduced mass simplifies calculations by treating the system as a single particle.

• Formula:

• Used in atomic, molecular, and celestial mechanics.

Velocity and Acceleration of the Center of Mass

• Velocity:

• Acceleration:

• Total momentum and force are sums of individual momenta and forces.

Application: Particle Physics

• In collisions, the center of mass frame is often used to analyze particle interactions.

• Colliding particles have equal and opposite momenta in the center of mass frame.

Application: Rockets

Rocket Propulsion and Momentum

Rocket motion is governed by the conservation of momentum as mass is ejected as propellant.

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• As fuel is expelled, the rocket's mass decreases and its velocity increases.

Impulse and Thrust

• Impulse formula:

• Thrust:

• Thrust has units of force (Newtons).

Rocket Equation

• Change in velocity:

• = effective exhaust velocity, = initial mass, = final mass

• Fuel requirements grow exponentially with desired .

Angular Momentum

Definition and Units

Angular momentum is the rotational analog of linear momentum, describing the quantity of rotation of an object.

• For a rigid body:

• SI unit: kg·m2/s

• For a point particle:

Angular Momentum for a Point Particle

• is the angle between the position vector and the momentum vector.

• The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.

Newton's Second Law for Rotational Motion

• Rate of change of angular momentum:

• Torque:

• Compare with linear form:

Conservation of Angular Momentum

• If the net external torque on a system is zero, angular momentum is conserved.

• Systems that change shape (e.g., a spinning figure skater pulling in arms) will change angular speed to conserve angular momentum.

• In rotational collisions, angular momentum is also conserved.

• Gyroscopes are stable due to conservation of angular momentum.

Rotational Work and Power

• Work done by torque:

• Power produced by torque:

• Analogous to linear work and power: ,

Vector Nature of Rotational Motion

• The direction of the angular velocity vector is along the axis of rotation.

• Right-hand rule determines the sign of angular velocity and torque vectors.

Examples

• Merry-Go-Round: A girl of mass 50 kg runs at 2 m/s to jump onto a stationary merry-go-round of radius 5 m. Her angular momentum is .

• Vertical Bar: A bar of length and mass attached to a hinge is released; find the speed of the end when it hits the ground using conservation of energy and angular momentum.

• Sphere and Bullet: A bullet of mass and velocity is fired into a sphere of mass and radius suspended by a string; find the angular velocity after collision.

Additional info:

• Special relativity is required for momentum and energy calculations at speeds close to the speed of light.

• Reduced mass is especially useful in atomic and celestial mechanics.

• Gyroscopes and spinning objects demonstrate conservation of angular momentum in practical applications.