In perfectly elastic collision in one dimension between two bodies the magnitudevelocity of each body

In a perfectly elastic collision involving two bodies in one dimension, both momentum and kinetic energy are conserved. This means that not only do the total momentum and total kinetic energy before the collision equal those after the collision, but the individual velocities of the bodies change in a specific way based on their initial velocities and masses. Let’s break this down step by step.

Understanding Momentum and Kinetic Energy Conservation

In any collision, the principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:

• Before Collision: \( m_1 v_{1i} + m_2 v_{2i} \)
• After Collision: \( m_1 v_{1f} + m_2 v_{2f} \)

Here, \( m_1 \) and \( m_2 \) are the masses of the two bodies, \( v_{1i} \) and \( v_{2i} \) are their initial velocities, and \( v_{1f} \) and \( v_{2f} \) are their final velocities after the collision.

Applying the Conservation Laws

For perfectly elastic collisions, we also have the conservation of kinetic energy, which can be expressed as:

• Before Collision: \( \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 \)
• After Collision: \( \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \)

By setting these two equations equal to each other, we can derive the final velocities of both bodies after the collision.

Deriving Final Velocities

To find the final velocities, we can use the following formulas derived from the conservation laws:

• Final velocity of body 1: \( v_{1f} = \frac{(m_1 - m_2)v_{1i} + 2m_2v_{2i}}{m_1 + m_2} \)
• Final velocity of body 2: \( v_{2f} = \frac{(m_2 - m_1)v_{2i} + 2m_1v_{1i}}{m_1 + m_2} \)

These equations show how the final velocities depend on the initial velocities and masses of the two bodies involved in the collision.

Example Scenario

Let’s consider a practical example. Suppose we have two bodies: body 1 with a mass of 2 kg moving at 3 m/s, and body 2 with a mass of 1 kg moving at -1 m/s (the negative sign indicates it’s moving in the opposite direction).

Using the formulas:

• For body 1:

\( v_{1f} = \frac{(2 - 1) \cdot 3 + 2 \cdot 1 \cdot (-1)}{2 + 1} = \frac{3 - 2}{3} = \frac{1}{3} \, \text{m/s} \)

• For body 2:

\( v_{2f} = \frac{(1 - 2) \cdot (-1) + 2 \cdot 2 \cdot 3}{2 + 1} = \frac{1 + 12}{3} = \frac{13}{3} \, \text{m/s} \)

After the collision, body 1 moves at approximately 0.33 m/s, while body 2 moves at approximately 4.33 m/s. This example illustrates how the velocities change due to the collision while conserving both momentum and kinetic energy.

Final Thoughts

In summary, perfectly elastic collisions in one dimension are fascinating because they allow us to see the interplay between momentum and kinetic energy. By applying the conservation laws, we can predict the final velocities of the bodies involved, which can be very useful in various fields, from physics to engineering. Understanding these principles not only deepens your grasp of mechanics but also enhances your problem-solving skills in real-world applications.