Describe the types of collisions in one dimension.

There are two types of collisions in one dimension: elastic and inelastic.

In an elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum before the collision is equal to the total momentum after the collision, and the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically, this can be expressed as:

m1v1i + m2v2i = m1v1f + m2v2f (conservation of momentum)
1/2m1v1i^2 + 1/2m2v2i^2 = 1/2m1v1f^2 + 1/2m2v2f^2 (conservation of kinetic energy)

where m1 and m2 are the masses of the two objects, v1i and v2i are their initial velocities, and v1f and v2f are their final velocities.

In an inelastic collision, only momentum is conserved, while kinetic energy is not. This means that the total momentum before the collision is equal to the total momentum after the collision, but the total kinetic energy before the collision is not equal to the total kinetic energy after the collision. Mathematically, this can be expressed as:

m1v1i + m2v2i = (m1 + m2)vf (conservation of momentum)

where vf is the final velocity of the two objects after the collision.

In both types of collisions, the coefficient of restitution (e) can be used to determine the relative velocities of the objects before and after the collision. The coefficient of restitution is defined as the ratio of the relative velocity of separation to the relative velocity of approach, and is given by:

e = (v2f - v1f) / (v1i - v2i)