A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if the collision is elastic.

Verified step by step guidance

1. Step 1: Understand the problem. This is a two-object collision problem where the collision is elastic. In an elastic collision, both momentum and kinetic energy are conserved. The goal is to find the final velocities of both objects after the collision.
2. Step 2: Write the equation for conservation of momentum. The total momentum before the collision equals the total momentum after the collision. The equation is: $m$₁v₁i + m₂v₂i = m₁v₁f + m₂v₂f, where $m$₁ and $m$₂ are the masses of the objects, $v$₁i and $v$₂i are their initial velocities, and $v$₁f and $v$₂f are their final velocities.
3. Step 3: Write the equation for conservation of kinetic energy. Since the collision is elastic, the total kinetic energy before the collision equals the total kinetic energy after the collision. The equation is: $1/2m$₁v₁i² + 1/2 m₂v₂i² = 1/2 m₁v₁f² + 1/2 m₂v₂f². Simplify this equation by canceling out the 1/2 factor.
4. Step 4: Solve the system of equations. You now have two equations: one from momentum conservation and one from kinetic energy conservation. Substitute the known values ($m$₁ = 5.5, $m$₂ = 8.0, $v$₁i = 6.5, $v$₂i = -4.0) into the equations. Solve for the two unknowns, $v$₁f and $v$₂f, using algebraic methods or substitution.
5. Step 5: Interpret the results. Once you solve for $v$₁f and $v$₂f, check that they satisfy both the momentum and kinetic energy conservation equations. These final velocities represent the speeds and directions of the objects after the elastic collision.

Key Concepts

Conservation of Momentum

In an isolated system, the total momentum before a collision is equal to the total momentum after the collision. This principle is crucial for analyzing collisions, as it allows us to set up equations based on the masses and velocities of the colliding objects to find their final velocities.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. This means that not only do the objects bounce off each other without losing energy, but the total kinetic energy before and after the collision remains the same, which is essential for solving problems involving elastic collisions.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In elastic collisions, the conservation of kinetic energy must be considered alongside momentum to determine the final velocities of the colliding objects.