Collisions in One Dimension

In this simulation, you get to collide two objects, and investigate whether momentum and/or kinetic energy are conserved in the collision process. To keep things simple, we'll confine ourselves to collisions along a single line - these will be one-dimensional collisions, in other words.

Momentum

Momentum is a vector - an object's momentum has the same direction as its velocity. Momentum is, in fact, the object's mass (m) multiplied by its velocity (v). In equation form, we get:

p = mv

The law of conservation of momentum states that, as long as no net external force acts on a system, the system's momentum is conserved. In our case, the system of two objects does not experience a net external force, so we do expect the system's momentum to be the same before and after the collision.

m₁ v_1i + m₂ v_2i = m₁ v_1f + m₂ v_2f

where 1 and 2 denote the two objects, and f represents values after the collision (final) and i represents values before the collision (initial). This is a vector equation. In one dimension, then, we need to use the appropriate plus or minus sign for each of the velocities.

Kinetic energy

In kinetic energy, we have another way to combine m (mass) and v, although kinetic energy is a scalar, so the v in this case is the speed, which is the magnitude of the velocity. The equation for kinetic energy is:

E_K = (1/2) mv²

Total energy in a closed system must always be conserved, but this does not mean that kinetic energy must be conserved - in a coliision, some kinetic energy is generally transformed into other forms of energy, such as thermal energy or sound energy. Kinetic energy is generally only the same before and after a collision when the collision is perfectly elastic.

Elasticity

The elasticity of a collision is defined as, essentially, the ratio of the relative velocity of the two objects after the collision to the negative of the relative velocity before the collision. In equation form, we can write this as:

k = (v_2f - v_1f) / (v_1i - v_2i),

where 1 and 2 denote the two objects, and f represents values after the collision (final) and i represents values before the collision (initial).

Special cases include the completely inelastic case, in which the objects stick together after the collision, and k = 0, as well as the completely elastic case, in which kinetic energy is conserved, and in which k = 1.