0.24 Phy1215: energy -- elastic and inelastic collisions in two  (Page 3/9)

What do the equations imply?

The first two equations based on momentum in Figure 1 require that the combined momentum of the two objects be the same, along each axis, beforeand after the collision. Thus, one of the equations deals with momentum along the horizontal axis and the other equation deals with momentum along thevertical axis.

The terms in the two equations are:

• m1 and m2 represent the masses of object 1 and object 2.
• u1x and u2x represent the components of the velocities of the two objects along the horizontal or x axis before the collision.
• u1y and u2y represent the components of the velocities of the two objects along the vertical or y axis before the collision.
• v1x and v2x represent the components of the velocities of the two objects along the horizontal or x axis after the collision.
• v1y and v2y represent the components of the velocities of the two objects along the vertical or y axis after the collision.

The third or energy equation

As indicated in Figure 1 , this equation can only be used for the case of an elastic collision. This equation requires that the total kinetic energy of thetwo objects before the collision be equal to the total kinetic energy of the two objects after the collision.

In addition to the masses described above, this equation introduces the following terms:

• u1 and u2 represent the magnitudes of the velocities of the two objects before the collision.
• v1 and v2 represent the magnitudes of the velocities of the two objects after the collision.

Definition of the angles

In order to compute the horizontal and vertical components of the velocities before and after the equations, you must know the directions in which theobjects are moving before and after the collision. Those directions appear as angles in Figure 1 where

• a1 and a2 are angles measured counter clockwise relative to the horizontal axis that represent the direction of travel of each of theobjects respectively before the collision.
• b1 and b2 are angles measured counter clockwise relative to the horizontal axis that represent the direction of travel of each of theobjects respectively after the collision.

Ten variables

As you can see in Figure 1 , these equations involve ten variables. That means that in order for a solution to be possible, the values for eight of thevariables must be know for an inelastic collision, and the values for seven of the variables must be known for an elastic collision.

Not conceptually difficult

If you believe in the laws of conservation of momentum and conservation of energy on which these three equations are based, the solution to collisionproblems is not conceptually difficult.

However, depending on which variables are known, which are unknown, and whether the collision is elastic, inelastic, or perfectly inelastic, you cancome up with equations that are difficult to solve from an algebra/trigonometry viewpoint.

Axis rotation

For the case where none of the given directions are along the x- axis or the y-axis, you can sometimes simplify the algebraic/trigonometric problem byrotating the axis so as to place one of those directions along the x-axis or the y-axis. This will often cause one or more terms in the set of equations to go tozero, thus simplifying the solution to the set of equations.