Elastic And Inelastic Collisions

Step 1: Apply conservation of momentum.
Before collision: After inelastic collision (masses stick together): total mass = , and velocity : Step 2: Apply conservation of energy after collision.
Kinetic energy after collision is converted into spring potential energy at maximum compression: Substitute : Step 3: Final Answer.
Maximum compression of the spring is

Step 1: Conservation of momentum.
For an elastic collision, momentum is conserved. The total momentum before and after the collision is equal: Simplifying, we get Step 2: Conservation of kinetic energy.
For an elastic collision, the total kinetic energy is also conserved: Simplifying, we get Step 3: Solve the system of equations.
From equation (1), Substitute this into equation (2): Expanding: Simplifying: Thus, Step 4: Final result.
Therefore, . Final Answer: The value of is .