An elastic collision is a special type of interaction between two objects where both total momentum and kinetic energy are conserved.

What is an Elastic Collision?

In physics, momentum is the mass of an object multiplied by its velocity. Kinetic energy is the energy an object has because it is moving. In an elastic collision, objects bounce off each other perfectly without losing energy to heat or sound. Think of two billiard balls hitting each other; they move apart immediately without getting stuck or deformed.

Conservation Laws

To find the final velocities, we use two main rules. The first rule is the Law of Conservation of Momentum. This means the total momentum before the collision equals the total momentum after the collision. The second rule is the Law of Conservation of Kinetic Energy. This means the total energy of motion is the same before and after the bounce.

The Mathematical Steps

Let two objects have masses m1 and m2, and initial velocities u1 and u2. After the collision, their final velocities become v1 and v2.

• Write the momentum equation: m1u1 + m2u2 = m1v1 + m2v2
• Write the kinetic energy equation: ½m1u1² + ½m2u2² = ½m1v1² + ½m2v2²
• Combine these two equations by dividing the kinetic energy equation by the momentum equation.
• This simplifies to: u1 + v1 = u2 + v2.

By rearranging this simple equation, we can substitute the value of v2 into the momentum equation to solve for v1, and vice versa.

Final Velocity Formulas

After solving the algebra, we get two key formulas for the final velocities:

• v1 = [(m1 - m2) / (m1 + m2)]u1 + [2m2 / (m1 + m2)]u2
• v2 = [2m1 / (m1 + m2)]u1 + [(m2 - m1) / (m1 + m2)]u2

These formulas allow you to predict exactly how fast objects will move after they collide, provided you know their starting mass and speed.