Final answer:
In an elastic head-on collision between two identical billiard balls, their final velocities are switched. The principles of conservation of momentum and kinetic energy are used to derive the final velocities after the collision.
Step-by-step explanation:
To find the final velocities of two billiard balls that undergo an elastic, head-on collision, we can employ the principles of conservation of momentum and conservation of kinetic energy. For an elastic collision involving two objects moving along the same line, the two conservation laws lead to two equations with the final velocities as the unknowns.
The conservation of momentum gives us:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
The conservation of kinetic energy gives us:
0.5 * m1 * v1_initial^2 + 0.5 * m2 * v2_initial^2 = 0.5 * m1 * v1_final^2 + 0.5 * m2 * v2_final^2
For two identical balls, m1 = m2, and simplifying the equations gives:
v1_initial + v2_initial = v1_final + v2_final,
v1_initial^2 + v2_initial^2 = v1_final^2 + v2_final^2.
Solving these equations with the given initial velocities (2.00 m/s and 4.00 m/s), we find that after an elastic collision:
v1_final = the initial velocity of the second ball,
v2_final = the initial velocity of the first ball.
Without knowing which ball has which initial velocity, we can say that their final velocities will simply be switched.