Elastic collisions in one dimension are fascinating events where objects collide and bounce off each other without losing energy. These collisions follow strict rules of physics, conserving both momentum and kinetic energy throughout the entire process.

Understanding elastic collisions helps us grasp how objects interact in idealized scenarios. We'll explore how to calculate final velocities, analyze energy conservation, and solve problems involving these perfect bounces between objects.

Elastic Collisions in One Dimension

Elastic collisions in one dimension

• Elastic collision involves two objects moving along a straight line (one-dimensional motion) where the total kinetic energy and momentum are conserved
• Objects make contact during the collision and continue moving along the same line after the collision
• No energy is lost or dissipated as heat, sound, or deformation during an elastic collision (pool balls, air track gliders)
• Momentum and kinetic energy of individual objects may change, but the total momentum and kinetic energy of the system remain constant
• A perfectly elastic collision is an idealized scenario where the coefficient of restitution is exactly 1

Final velocities after elastic collisions

• Calculate final velocities using conservation of momentum and kinetic energy equations
• Conservation of momentum: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$
  • $m_1$, $m_2$ represent masses of the objects
  • $v_1$, $v_2$ represent initial velocities before collision
  • $v_1'$, $v_2'$ represent final velocities after collision
• Conservation of kinetic energy: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$
• Solve the system of equations simultaneously to determine the final velocities of the objects after the elastic collision
• The relative velocity between the objects is important in determining the outcome of the collision

Conservation of internal kinetic energy

• Total kinetic energy remains constant before and after an elastic collision
• Kinetic energy of individual objects may change, but the sum of their kinetic energies is conserved
• Analyze the collision using the conservation of kinetic energy equation: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$
• Combine with the conservation of momentum equation to solve for the final velocities
• Internal kinetic energy refers to the kinetic energy within the system of colliding objects (billiard balls, colliding carts on a track)

Center of mass in elastic collisions

• The center of mass of the system continues to move at a constant velocity during and after the collision
• In a closed system, the motion of the center of mass is not affected by internal forces during the collision
• The velocity of the center of mass can be calculated using the masses and velocities of the colliding objects
• Newton's cradle demonstrates the conservation of momentum and energy through a series of elastic collisions

Problem-solving for elastic collisions

1. Identify given information:

   • Masses of the objects ($m_1$ and $m_2$)
   • Initial velocities of the objects ($v_1$ and $v_2$)

2. Write conservation of momentum equation: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

3. Write conservation of kinetic energy equation: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$

4. Solve the system of equations for the final velocities ($v_1'$ and $v_2'$):

   • Substitute known values into the equations
   • Manipulate equations to isolate unknown variables
   • Solve for the final velocities algebraically or using substitution

5. Check solutions by plugging final velocities back into original equations to ensure conservation of momentum and kinetic energy (colliding marbles, air hockey pucks)