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.

m_{1} v_{1i} + m_{2} v_{2i} = m_{1}
v_{1f} + m_{2} 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) mv^{2 }

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.

]]>- Start with the initial values of all the sliders, and run the simulation by pressing the Play button. This is a relatively simple collision, a perfectly elastic collision between objects of equal mass. See if you can find the three momentum and the three kinetic energy values after the collision. Note that you can enter these one at a time, and use the "Check Answer" button to check each individual answer.
- Reset the simulation (press the "Reset Simulation" button), and then set the mass of ball 1 to 3 kg, and the elasticity to 0. This gives what we call a completely inelastic collision, in which the objects stick together after the collision. See if you can predict the answers, for the momentum and kinetic energy afterwards, before running the simulation.
- Reset the simulation again, and then set the mass of ball 1 to 3 kg. This should give you an elastic collision between objects with equal-and-opposite velocities before the collision, and a mass ratio of 3. This is a special case. See if you can predict the answers, for the momentum and kinetic energy afterwards, before running the simulation - make sure you Play it to see the result, though.
- Adjust the sliders several times, running through various combinations of parameters. Under what conditions is the total momentum conserved in the collision? Under what conditions is the total kinetic energy conserved in the collision?
- Finally, try a collision in which the elasticity is 0.5. Using the momentum conservation equation, and the elasticity equation, you should be able to solve for the momentum and kinetic energy after the collision. See if you can do this.