A \(1 \mathrm{~kg}\) mass strikes a \(2 \mathrm{~kg}\) mass in an inelastic collision on a horizontal frictionless surface. The \(1 \mathrm{~kg}\) mass is traveling at \(1 \mathrm{~m} / \mathrm{s}\) in the positive direction and the \(2 \mathrm{~kg}\) mass is traveling at \(1 \mathrm{~m} / \mathrm{s}\) in the negative direction. The final velocity of the masses is (A) \(+1 \mathrm{~m} / \mathrm{s}\) (B) \(+1 / 2 \mathrm{~m} / \mathrm{s}\) (C) \(-1 \mathrm{~m} / \mathrm{s}\) (D) \(-2 / 3 \mathrm{~m} / \mathrm{s}\) (E) \(-1 / 3 \mathrm{~m} / \mathrm{s}\)

Understanding the principle of momentum conservation is crucial when analyzing collisions. In physics, momentum refers to the product of an object's mass and velocity, and it is a vector quantity, which means it has both magnitude and direction. The Law of Conservation of Momentum states that in a closed system, where no external forces are acting, the total momentum before an event is equal to the total momentum after the event.

This principle is particularly useful in collision problems. When two bodies collide, as long as no external forces intervene, the total momentum of the bodies before the collision will be the same after the collision, regardless of the type of collision—be it elastic or inelastic.

In our example, we apply this law to determine the final velocity after an inelastic collision. Despite the collision being inelastic, meaning kinetic energy is not conserved, momentum conservation holds true, as there are no external forces acting on the system. This allows us to solve the problem using a simple, yet powerful equation that relates the initial momenta of both masses to their combined momenta after the collision.

Collision theory is a concept that helps explain the dynamics of a collision between two or more bodies. The theory delineates between elastic and inelastic collisions. In an elastic collision, both momentum and kinetic energy are conserved. Conversely, in an inelastic collision, while momentum is conserved, kinetic energy is not. Some of the kinetic energy is transformed into other forms of energy, such as sound or thermal energy. This is often observable in real-world impacts where deformation or heat generation occurs.

Our given problem exemplifies an inelastic collision, where two objects stick together after colliding. The singular final velocity of the combined mass post-collision is a hallmark of an inelastic collision. Knowing the type of collision allows us to apply the correct principles to solve for the final velocities. Even though the calculation steps assume that objects adhere to each other, in a broader sense, inelastic collision also includes scenarios where some kinetic energy is lost but the bodies do not necessarily merge.

The final velocity calculation in collision problems is fundamental. After applying the law of conservation of momentum, we obtain an equation that relates the masses and velocities of colliding bodies before and after the collision. From this equation, we can isolate and solve for the final velocity.

As seen in our exercise, this process begins with calculating the initial momentum using the given masses and velocities. Afterwards, due to momentum conservation in an inelastic collision, we equate the sum of the initial momenta to the final combined mass times the final velocity. The solution involves algebraic manipulation to isolate the final velocity, which is what we're looking to find. This example demonstrates the simplicity of final velocity calculations even in concepts that might appear complex at first glance.

To enhance understanding, it's helpful to present such problems with their real-world applications. For instance, one could tie the concept to how crumple zones in cars are designed to absorb the impact energy, effectively reducing the velocity of the vehicles—demonstrating the conservation of momentum and energy dissipation during inelastic collisions.