Chapter 8 LINEAR MOMENTUM AND COLLISIONS

• Linear Momentum

• Momentum and Newton’s Second Law

• Impulse

• Conservation of Linear Momentum

• Inelastic Collisions

• Elastic Collisions

• Center of Mass

• Systems with Changing Mass: Rocket Propulsion

Linear Momentum

Change in momentum:

(a) mv

(b) 2mv

Momentum and Newton’s Second Law Newton’s

second law, as we wrote it before:

Impulse Impulse is a vector, in the same direction as the average force.

Therefore, the same change in momentum may be produced by a large force

acting for a short time, or by a smaller force acting for a longer time.

Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum:

Conservation of Linear Momentum

An example of internal forces moving components of a system:

Inelastic Collisions Collision: two objects striking one another Time of collision is short enough that external forces

may be ignored

Inelastic collision: momentum is conserved but kinetic energy is not Completely inelastic

collision: objects stick together afterwards

Inelastic Collisions A completely inelastic collision:

Inelastic Collisions

Solving for the final momentum in terms of the initial momenta and masses:

Ballistic pendulum:

the height h can be found using conservation of mechanical energy after the object is embedded in the block.

For collisions in two dimensions, conservation of momentum is applied separately along each axis:

Elastic Collisions In elastic collisions, both kinetic energy and momentum are conserved.

One-dimensional elastic collision:

Elastic Collisions We have two equations

(conservation of momentum and conservation of

kinetic energy) and two unknowns (the final speeds). Solving for the final speeds:

Elastic Collisions Two-dimensional collisions can only be solved if some of the final

information is known, such as the final velocity of one object:

Center of Mass The center of mass of a system is the point where the system

can be balanced in a uniform gravitational field.

The center of mass need not be within the object:

The total mass multiplied by the acceleration of the center of mass is equal to the net external force:

The center of mass accelerates just as though it were

a point particle of mass M acted on by

Systems with Changing Mass: Rocket Propulsion If a mass of fuel Δm is

ejected from a rocket with speed v, the change in momentum of the rocket is:

The force, or thrust, is

Summary

• Momentum is conserved if the net external force is zero

• Internal forces within a system always sum to zero

• In collision, assume external forces can be ignored

• Inelastic collision: kinetic energy is not conserved

• Completely inelastic collision: the objects stick together afterward

• A one-dimensional collision takes place along a line

• In two dimensions, conservation of momentum is applied separately to each

• Elastic collision: kinetic energy is conserved

• Center of mass:

• Motion of center of mass:

• Rocket propulsion:

An elastic one-dimensional two-object collision. Momentum and internal kinetic energy are conserved.

• An inelastic one-dimensional two-object collision. Momentum is conserved, but internal kinetic energy is not conserved.

(a) Two objects of equal mass initially head directly toward one another at the same speed.

(b) The objects stick together (a perfectly inelastic collision), and so their final velocity is zero. The internal kinetic energy of the system changes in any inelastic collision and is reduced to zero in this example.

• A two-dimensional collision with the coordinate system chosen so that m2 is initially at rest and v1 is parallel to the x -axis. This coordinate system is sometimes called the laboratory coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it to determine the particles that make-up the target and how they are bound together. The particles may not be observed directly, but their initial and final velocities are.