Final answer:
The correct formula to use for calculating the final velocities after a collision, assuming no external forces, is m1v1 + m2v2 = m1v1f + m2v2f, applicable for both elastic and inelastic collisions.
Step-by-step explanation:
When two carts, whether of equal or different masses, collide and move separately after the collision, the conservation of momentum is applicable to calculate the final velocities of the carts. The correct formula to use is m1v1 + m2v2 = m1v1f + m2v2f, where m1 and m2 are the masses of the carts, v1 and v2 are the initial velocities, and v1f and v2f are the final velocities of cart 1 and cart 2, respectively. This principle holds as long as no external forces act on the system, such as friction. Looking at different types of collisions, if it's an inelastic collision, the two objects stick together after the impact. In contrast, in an elastic collision, the objects bounce back separately while conserving both momentum and kinetic energy.
This formula is based on the law of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision. In this formula, m1 and m2 represent the masses of the two carts, v1 and v2 represent their initial velocities, and v1f and v2f represent their final velocities.
By using this formula, you can calculate the final velocities of the carts based on their initial velocities and masses.