## Theory

When two objects collide and there is no external force acts upon them, the vector sum of their momenta, * ***p** = m **v**, is always conserved. This means the total momentum after the collision (f = final) is the same as that before the collision (i = initial):

[sum of *p*]_{i} = [sum of *p*]_{f}
*p*_{1i} + p_{2i} = p_{1f} + p_{2f} .

(Note: the indices "1" and "2" refer to the object #1 and object #2, respectively). In other words, the ratio of the total momentum after collision to that before collision is equal to one: [sum of *p*]_{f} / [sum of *p*]_{i} = 1. All kinds of collision, whether it is perfectly elastic, partially elastic, or totally inelastic, satisfy the conservation of momentum as long as there is no external force acting on the colliding objects during the collision.
If the forces acting during the collision are conservative, then mechanical energy is conserved as well. If the objects moves along horizontal, only the kinetic energy, *KE = ½ m v*^{2} plays a role. Therefore

[sum of *KE*]_{i} = [sum of *KE*]_{f}
½ *m*_{1} v^{2}_{1i} + ½ m_{2} v^{2}_{2i} = ½ m_{1} v^{2}_{1f} + ½ m_{2} v^{2}_{2f} .

In other words, the ratio of the total kinetic energies after collision to that before collision is equal to one: [sum of *KE*]_{f} / [sum of *KE*]_{i} = = 1. The conservation of mechanical energy is satisfied only by perfectly elastic collisions. In partially elastic collisions, and in totally inelastic collisions (in which the two objects stick together after the collision: *v*_{1f} = v_{2f}), a part of energy is lost during the collision ("lost" means converted into other forms of energy, such as heat or sound, which are not mechanical), and therefore the ratio of the total kinetic energy after the collision to that before the collision is substantially less than one.
In this experiment, we will use an air track as an essentially frictionless surface. (An air track is an aluminum surface with many tiny holes punched regularly on it, such that when sufficient amount of air is blown through the holes continuously by means of an electric air blower, an object sliding on this thin cushion of air is considered to be moving without friction). Since all motions are one dimensional, the momenta can be added algebraically with appropriate positive or negative signs to incorporate the direction of motion. Furthermore, in order to simplify the analysis, the 2nd glider (*m*_{2}) will be held to stay still (*v*_{2i} = 0) before the collision, such that only the 1st glider (m_{1}) will have an initial velocity *v*_{1i} directed toward the stationary 2nd glider. For the Elastic Collision, the masses of the gliders are chosen to be almost exactly equal to each other (m_{1} __~__ m_{2}), such that after collision, both laws of conservation of momentum and kinetic energy give rise to: the 1st glider comes almost to a full stop (v_{1f} __~__ 0), and the 2nd glider takes off with the velocity almost equal to the initial velocity of the 1st glider (v_{2f} __~__ v_{1i}). (See the following figures).

**ELASTIC COLLISION:**

**TOTALLY INELASTIC COLLISION:**

The x-component of the conservation of momentum (for all collisions in this experiment) is then given as

*m*_{1} (+v_{1i} ) = m_{1} (+v_{1f} ) + m_{2} (+v_{2f} ),
and the conservation of mechanical energy (only for perfectly elastic collision in this experiment) is given as
½ *m*_{1} v^{2}_{1i} = ½ m_{1} v^{2}_{1f} + ½ m_{2} v^{2}_{2f} .
The instantaneous velocities v's will be determined in an approximate manner, by means of photogates, as *v = delta x/ delta t* with * delta t* a very small number. The numerator *delta x* is the width of a small flag mounted vertically at the top of the glider, whereas *delta t* is the time interval during which the flag blocks the photogate as the glider is sliding along the track.