Solving a Collision Problem Using Conservation of Momentum

In this article, we will delve into a practical example of a collision problem, illustrating the application of the principle of conservation of momentum. Once we understand and master this fundamental concept, we can apply it to various physics problems, including those related to sports, automotive safety, and even celestial mechanics.

Problem Statement

A ball A (mass 0.1 kg) moves with a velocity of 6 m/s and collides with another ball B (mass 0.2 kg) that is initially at rest. After the collision, ball A rebounds with a velocity of 2 m/s in the opposite direction. The question is: what is the velocity of ball B after the collision?

Principle of Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In this context, the collision between balls A and B is considered the system under study.

Step-by-Step Solution

Step 1: Set Up the Problem Using the Conservation of Momentum

We can express the conservation of momentum as:

Initial momentum Final momentum

Where momentum (p) is given by the formula:

momentum mass × velocity

Step 2: Calculate the Initial Momentum

The initial momenta of balls A and B are:

Initial momentum of ball A: m_Au_A

Initial momentum of ball B: m_Bu_B

Given:

Initial momentum:

Initial momentum of ball A 0.1 kg × 6 m/s 0.6 kg·m/s

Initial momentum of ball B 0.2 kg × 0 m/s 0 kg·m/s

Total initial momentum 0.6 kg·m/s 0 kg·m/s 0.6 kg·m/s

Step 3: Calculate the Final Momentum

The final momenta of balls A and B are:

Final momentum of ball A: m_Av_A

Final momentum of ball B: m_Bv_B

Given:

Final momentum of ball A 0.1 kg × -2 m/s -0.2 kg·m/s

Total final momentum -0.2 kg·m/s 0.2 kg·m/s v_B

Step 4: Apply the Conservation of Momentum

Setting initial momentum equal to final momentum:

0.6 kg·m/s -0.2 kg·m/s 0.2 kg·m/s v_B

Solving for v_B:

0.6 0.2 0.2 v_B

0.8 0.2 v_B

v_B 4 m/s

Conclusion

The velocity of ball B after the collision is 4 m/s in the same direction as the initial velocity of ball A.

Keywords: conservation of momentum, collision problem, physical science