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 objects stick together.

Verified step by step guidance

1. Step 1: Identify the type of collision. Since the objects stick together after the collision, this is an inelastic collision. In such cases, momentum is conserved, but kinetic energy is not.
2. Step 2: Write the equation for the conservation of momentum. The total momentum before the collision equals the total momentum after the collision. Mathematically, this is expressed as: $m$₁v₁i + m₂v₂i = (m₁ + m₂)v_f, where $m$₁ and $m$₂ are the masses of the two objects, $v$₁i and $v$₂i are their initial velocities, and $v$_f is their final velocity after sticking together.
3. Step 3: Substitute the given values into the momentum conservation equation. Use $m$₁ = 5.5 kg, $v$₁i = 6.5 m/s, $m$₂ = 8.0 kg, and $v$₂i = -4.0 m/s (negative because it is in the opposite direction). The equation becomes: $(5.5)(6.5)\; +\; (8.0)(-4.0)\; =\; (5.5\; +\; 8.0)v$_f.
4. Step 4: Simplify the left-hand side of the equation to calculate the total initial momentum. Then, divide by the combined mass ($5.5\; +\; 8.0$) to solve for $v$_f, the final velocity of the combined object.
5. Step 5: Interpret the result. The final velocity $v$_f will indicate the direction and speed of the combined object after the collision. A positive value means it moves in the +𝓍 direction, while a negative value means it moves in the ―𝓍 direction.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In collisions, the momentum before the collision equals the momentum after the collision. This principle is crucial for solving problems involving collisions, as it allows us to relate the velocities and masses of the colliding objects.

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic. In elastic collisions, both momentum and kinetic energy are conserved, while in inelastic collisions, momentum is conserved but kinetic energy is not. The scenario described in the question involves an inelastic collision, where the two objects stick together after the collision, allowing us to use the conservation of momentum to find their final velocity.

Calculating Final Velocity

To find the final velocity of objects after a collision, we can use the formula derived from the conservation of momentum: (m1 * v1 + m2 * v2) = (m1 + m2) * v_final. Here, m1 and m2 are the masses of the objects, and v1 and v2 are their initial velocities. This equation allows us to solve for the final velocity when the objects stick together, combining their masses and initial momenta.