Understanding the Concepts
* Perfectly Elastic Collision: In a perfectly elastic collision, kinetic energy is conserved. This means the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision.
* Momentum Conservation: In any collision, momentum is always conserved. This means the total momentum of the system before the collision is equal to the total momentum after the collision.
Let's break down the situation:
* Ball 1: Initial velocity = *v₁*
* Ball 2: Initial velocity = *v₂*
* Final Velocities:
* Ball 1: *v₁'*
* Ball 2: *v₂'*
Applying the Conservation Laws
1. Conservation of Momentum:
* m*v₁* + m*v₂* = m*v₁'* + m*v₂'*
* Since the masses are equal, we can simplify: v₁ + v₂ = v₁' + v₂'
2. Conservation of Kinetic Energy:
* (1/2)mv₁² + (1/2)mv₂² = (1/2)mv₁'² + (1/2)mv₂'²
* Again, simplifying because the masses are equal: v₁² + v₂² = v₁'² + v₂'²
Solving for the Final Velocities
We now have two equations and two unknowns (v₁' and v₂'). Here's how to solve:
1. Rearrange the momentum equation:
* v₁' = v₁ + v₂ - v₂'
2. Substitute this into the kinetic energy equation:
* (v₁ + v₂ - v₂')² + v₂'² = v₁² + v₂²
3. Expand and simplify:
* v₁² + 2v₁v₂ + v₂² - 2v₁v₂' - 2v₂v₂' + v₂'² + v₂'² = v₁² + v₂²
* 2v₂'² - 2v₁v₂' - 2v₂v₂' = 0
* v₂'² - (v₁ + v₂)v₂' = 0
4. Factor:
* v₂'(v₂' - (v₁ + v₂)) = 0
5. Solve for v₂':
* v₂' = 0 or v₂' = v₁ + v₂
6. Substitute these values back into the momentum equation to find v₁':
* If v₂' = 0, then v₁' = v₁ + v₂
* If v₂' = v₁ + v₂ , then v₁' = 0
Interpreting the Results
* Case 1: v₂' = 0, v₁' = v₁ + v₂ This means Ball 2 comes to a complete stop, and Ball 1 moves forward with the combined velocity of the two balls.
* Case 2: v₂' = v₁ + v₂, v₁' = 0 This means Ball 1 comes to a complete stop, and Ball 2 moves forward with the combined velocity of the two balls.
In summary: In a perfectly elastic head-on collision of two billiard balls of equal mass, one ball will come to a complete stop, and the other ball will move forward with the combined initial velocity of the two balls.
