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Momentum and collisions (chapter eight). Linear momentum and its conservation Impulse and momentum Collisions Two-dimensional collisions Center of mass Systems of particles. Linear Momentum. Linear momentum is defined as the mass times the velocity of an object:
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Momentum and collisions (chapter eight) • Linear momentum and its conservation • Impulse and momentum • Collisions • Two-dimensional collisions • Center of mass • Systems of particles.
Linear Momentum Linear momentum is defined as the mass times the velocity of an object: This is a vector equation - in terms of the Cartesian coordinates:
Linear momentum The time rate of change of momentum of a particle equals the net force acting on it
Momentum conservation Consider an isolated system of two-particles, interacting with each other. The force on object 1 exerted by object 2 is The reaction force to this is the force on 2 by 1 Since these make up an action-reaction pair ConcepTest
Momentum conservation • The result is a basic statement of the law of conservation of the total momentum for an isolated system. • We can use this result to solve many types of collision and interaction problems in which we can choose the system to include two (or more) interacting objects. • Momentum is a vector, so its conservation holds separately for each component
Examples Two carts (masses m1 and m2) are held together at rest on a frictionless cart with a spring between them. When released, what are their velocities? After Before m1 m2
Conservation of momentum • Examples • Demo • ConcepTest
Impulse and momentum The change in momentum of an object during a collision is caused by a net force. Integrating the differential for the momentum: This quantity (the net force integrated over time) is called impulse, and results in another statement of Newton:s 2nd law: The total impulse of the net force equals the change in momentum of the particle.
Impulse approximation The time average of the force is defined as So the impulse momentum theorem can be written In the impulse approximation, we assume one of the forces acting on the object is much larger than the others, but acts only for a small interval.
Collisions Use the impulse approximation - forces of collisions much larger, shorter duration than other forces on objects. Two extremes of collisions: Elastic - kinetic energy conserved (e.g. billiard balls) Perfectly inelastic - object stick together (maximum KE loss) Momentum is conserved in both
Collisions, cont. Momentum conservation Energy conservation • One dimensional elastic After lots of algebra
2-d Collisions Momentum conservation – vector equation – holds for each component For elastic collisions – conservation of kinetic energy (magnitude of the velocity) Example
Center of mass = the average position of a system’s mass in the case of discrete point objects for an extended object
The white part if your fingernail is called the lunia. • No U.S. President has had brown eyes. • In the year 2160, there will be two lunar eclipses and five solar eclipses. • Benjamin Franklin invented the rocking chair. • 3 out of 4 people make up 75% of the population.
Motion of a system of particles = the velocity of the center of mass of a system of particles is just in other words, the total momentum is just the total mass times the velocity of the center of mass. The acceleration of the COM is
Motion of a system of particles The forces Fi has terms that are internal and external, but the internal forces sum to zero by Newton’s 3rd law So the system moves like a particle of mass M located at the COM. If the net external force is zero, the total momentum of the system is conserved
The hardness of butter is directly proportional to the softness of the bread. • To steal ideas from one person is plagiarism; to steal from many is research. • For every action there is an equal and opposite criticism. • If the speed of light is 186,000 miles/sec., what's the speed of darkness? • The sooner you fall behind the more time you'll have to catch up.
