Lecture Outlines

Chapter 9

Physics: Principles and Problems

Chapter 9Linear Momentum and

Collisions

Units of Chapter 9

Linear Momentum

Impulse

Conservation of Linear Momentum

Elastic and Inelastic Collisions

Center of Mass

Jet Propulsion and Rockets

6.1 Linear Momentum

Definition of linear momentum:

The linear momentum of an object is the product of its mass and velocity.

Note that momentum is a vector—it has both a magnitude and a direction.

SI unit of momentum: kg • m/s. This unit has no special name.

6.1 Linear Momentum

For a system of objects, the total momentum is the vector sum of each.

6.1 Linear MomentumThe change in momentum is the difference between the momentum vectors.

6.1 Linear Momentum

If an object’s momentum changes, a force must have acted on it.

The net force is equal to the rate of change of the momentum.

6.2 Impulse

Impulse is the change in momentum:

Typically, the force varies during the collision.

6.2 Impulse

Actual contact times may be very short.

6.2 ImpulseWhen a moving object stops, its impulse depends only on its change in momentum. This can be accomplished by a large force acting for a short time, or a smaller force acting for a longer time.

6.2 Impulse

We understand this instinctively—we bend our knees when landing a jump; a “soft” catch (moving hands) is less painful than a “hard” one (fixed hands).

This is how airbags work—they slow down collisions considerably—and why cars are built with crumple zones.

6.3 Conservation of Linear Momentum

If there is no net force acting on a system, its total momentum cannot change.

This is the law of conservation of momentum.

If there are internal forces, the momenta of individual parts of the system can change, but the overall momentum stays the same.

6.3 Conservation of Linear Momentum

In this example, there is no external force, but the individual components of the system do change their momenta:

6.3 Conservation of Linear Momentum

Collisions happen quickly enough that any external forces can be ignored during the collision. Therefore, momentum is conserved during a collision.

6.4 Elastic and Inelastic Collisions

In an elastic collision, the total kinetic energy is conserved.

Total kinetic energy is not conserved in an inelastic collision.

6.4 Elastic and Inelastic Collisions

A completely inelastic collision is one where the objects stick together afterwards.

6.4 Elastic and Inelastic Collisions

The fraction of the total kinetic energy that is left after a completely inelastic collision can be shown to be:

6.4 Elastic and Inelastic Collisions

For an elastic collision, both the kinetic energy and the momentum are conserved:

6.4 Elastic and Inelastic Collisions

Collisions may take place with the two objects approaching each other, or with one overtaking the other.

6.5 Center of Mass

Definition of the center of mass:

The center of mass is the point at which all of the mass of an object or system may be considered to be concentrated, for the purposes of linear or translational motion only.

We can then use Newton’s second law for the motion of the center of mass:

6.5 Center of MassThe momentum of the center of mass does not change if there are no external forces on the system.

The location of the center of mass can be found:

This calculation is straightforward for a system of point particles, but for an extended object calculus is necessary.

6.5 Center of Mass

The center of mass of a flat object can be found by suspension.

6.5 Center of Mass

The center of mass may be located outside a solid object.

6.6 Jet Propulsion and Rockets

If you blow up a balloon and then let it go, it zigzags away from you as the air shoots out. This is an example of jet propulsion. The escaping air exerts a force on the balloon that pushes the balloon in the opposite direction.

Jet propulsion is another example of conservation of momentum.

6.6 Jet Propulsion and RocketsThis same phenomenon explains the recoil of a gun:

6.6 Jet Propulsion and Rockets

The thrust of a rocket works the same way.

6.6 Jet Propulsion and RocketsJet propulsion can be used to slow a rocket down as well as to speed it up; this involves the use of thrust reversers. This is done by commercial jetliners.

Summary of Chapter 6

Momentum of a point particle is defined as its mass multiplied by its velocity.

The momentum of a system of particles is the vector sum of the momenta of its components.

Newton’s second law:

Summary of Chapter 6

Impulse–momentum theorem:

In the absence of external forces, momentum is conserved.

Momentum is conserved during a collision.

Kinetic energy is also conserved in an elastic collision.

Summary of Chapter 6

The center of mass of an object is the point where all the mass may be considered to be concentrated.

Coordinates of the center of mass: