Conservation of Momentum in Collisions: Understanding the Final Velocity of a Ball and Block System

Conservation of momentum is a fundamental concept in physics, specifically in the study of collisions. This principle helps us understand how the motion and mass of objects change during collisions, whether elastic or inelastic. In this article, we explore the application of conservation of momentum in a specific scenario involving a ball striking and sticking to a block resting on a frictionless surface.

Scenario Overview

Consider a 50 kg ball moving at 20 m/s that strikes and sticks to a 70 kg block which is initially at rest on a frictionless surface. Our objective is to find the final velocity of the block and the ball after they collide and stick together.

Applying the Principle of Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant before and after a collision. This is mathematically expressed as:

m1v1 m2v2 m1vf m2vf

where:

m1 is the mass of the first object (the ball), m2 is the mass of the second object (the block), v1 is the initial velocity of the first object (the ball), v2 is the initial velocity of the second object (the block), vf is the final velocity of the combined system, which is the same for both objects since they stick together.

Calculation of the Final Velocity

Given:

m1 50 kg v1 20 m/s m2 70 kg v2 0 m/s (the block is at rest)

Using the principle of conservation of momentum:

m1v1 m2v2 (m1 m2)vf

Solving for vf:

50 times; 20 70 times; 0 (50 70) vf

1000 120 vf

vf 1000 / 120 ≈ 8.33 m/s

Therefore, the final velocity of the combined system (ball and block) after the collision is approximately 8.33 m/s.

Understanding the Scenario

This scenario represents an inelastic collision, where the objects stick together after the collision. In inelastic collisions, kinetic energy is not conserved, but momentum is conserved.

General Formula for Inelastic Collisions

The general formula for an inelastic collision, where two objects stick together, is given by:

m1v1i m2v2i (m1 m2)vf

Substituting the values from our scenario:

50 times; 20 70 times; 0 (50 70) vf

1000 120 vf

vf 1000 / 120 ≈ 8.33 m/s

Conclusion

In this system, the ball and the block, after striking and sticking together, move at a final velocity of approximately 8.33 m/s. Conservation of momentum is a powerful tool that allows us to solve such problems involving collisions, providing insights into the behavior of objects in motion.