In many situations, the force acting on an object is not constant, but changes with time. (Think about a person swinging a baseball bat). The magnitude of the force is zero at the instant right before it contacts the object. During contact, the net force rises to a maximum, and then returns to zero when the force leaves the object. The product of the average force and the time of contact is called the impulse of the force. Impulse is a vector quantity and has the same direction as the average force.

Impulse = F*t, where F is the average force exerted by an outside source, and t is in seconds.

Both mass and velocity play a role in how an object responds to a given impulse. These effects are included in the concept of linear momentum. Linear momentum is a quantity that points in the same direction as the velocity.

Linear Momentum = m*(Vf-Vi), the velocities are for the entire system, taking the final minus the initial to get the overall change in velocity.

When a net force acts on an object, the impulse of the net force is equal to the change in momentum of the object. Momentum is a vector and has a magnitude and direction.

F*t = m*(Vf-Vi)

*Figures 7.4, 7.5, 7.6

When two objects approach each other in midair, their collision is known as a "system". There are two types of forces acting on the system. Internal forces are forces that the objects within the system exert on each other. External forces are forces exerted on objects by agents that are external to the system. The internal forces always add together to equal zero, therefore, internal forces can be ignored. If the sum of external forces is zero, the system is known as an "isolated system".

The principle of conservation of linear momentum states that the total linear momentum of an isolated system remains constant. If the sum of the average external forces is zero, then the final and initial total momentum are equal. The total linear momentum may be conserved even when the kinetic energies of the individual parts of a system change. Internal forces cannot change the total linear momentum of an isolated system because the total linear momentum is conserved in the presence of such forces.

To get the equation for two objects colliding and conserving momentum, check the math notes to get the equations alone, and when they are added together. Also, with the assumption that the force applied by external sources = 0, you can state that:

Pf = Pi, where the momentum in the end is equal to that in the beginning, proving that momentum is conserved within a system.

When objects are atoms or subatomic particles, the total kinetic energy of the system is often conserved. The total kinetic energy of the particles before the collision equals the total kinetic energy of the particles after the collision, so that the kinetic energy gained by one particle is lost by another. Kinetic energy is mainly lost in two ways. It can either be converted into heat because of friction, or spent in creating permanent distortion or damage (like in a car crash).

Collisions are often classified according to whether the total kinetic energy changes during the collision. Elastic collisions are those in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision. An Inelastic collision is one in which the total kinetic energy of the system is not the same before and after the collision. If the objects stick together, the collision is said to be inelastic. In an elastic collision, a steel ball, for example, dropped onto a hard surface will bounce back to its original height. On the other hand, a completely deflated basketball that is dropped onto a hard surface will not bounce back. This is an inelastic collision.

To find the degree of the collision in a system, you can use this equation, solving for any unknowns:

KEf1 + KEf2 = KEi1 + KEi2, where KE = (1/2)m*v^2

Solving this will show you where the kinetic energy is gained, lost, or constant.

Collisions can occur in all dimensions, even in two or three. For a system consisting of two balls, external forces would include the weights of the balls, and the corresponding normal forces. Since a normal force balances each weight, the sum of the external forces is zero, and the total momentum of the system is conserved. Momentum is a vector quantity, however, and in two dimensions, the x and y components are conserved separately.

So to calculate the separate quantities, take the given quantity of a system (such as the velocity), and multiply it by either sin or cos to get your y and x components. because the collision happen in two dimensions, each component must be treated separately. The equation is shown in the math notes.