Apply conservation of momentum to elastic and inelastic collisions in one dimension, and distinguish them using kinetic energy.

What this dot point is asking

You need to apply the law of conservation of momentum to one-dimensional collisions, write a correct before-and-after momentum equation (with signs for direction), and use kinetic energy to decide whether a collision is elastic or inelastic.

Why momentum is conserved

Newton's third law says the forces two colliding bodies exert on each other are equal and opposite. During the collision these are the only significant forces (the collision is brief), so the impulse on one body is the exact negative of the impulse on the other. Equal and opposite impulses mean equal and opposite momentum changes, so the total momentum change of the system is zero.

The "isolated" condition matters. If an external force such as friction or an applied push acts during the interaction, momentum is not conserved for the bodies alone. In most SACE collision problems we treat the interaction as brief enough that external forces are negligible.

Elastic vs inelastic collisions

The dividing line is kinetic energy, not momentum (momentum is conserved in both).

To classify a collision, compute total $E_k = \tfrac{1}{2}mv^2$ before and after. If they match, it is elastic; if $E_k$ after is less, it is inelastic.

Solving a collision

1. Draw before and after, mark velocities with signs.
2. Write the momentum equation and substitute known values.
3. If the objects stick together (perfectly inelastic), use one final velocity $v$ for both.
4. To classify, compare total kinetic energy before and after.

Recoil and explosions

An "explosion" is the reverse of a perfectly inelastic collision: a system starts at rest (total momentum zero) and separates into pieces. Because total momentum stays zero, the pieces fly apart with equal and opposite momenta. This is how recoil works - a fired projectile and the gun gain momenta of equal magnitude in opposite directions.