Understanding Elastic Collisions in Physics: Key Concepts and Calculations

Elastic collisions are an important topic in physics, offering insights into the principles of conservation of momentum and energy. In an elastic head-on collision, two objects rebound from each other without any loss of kinetic energy. This principle can be demonstrated through a detailed analysis of an example involving two carts connected by a spring. Let's explore the scenario where a 0.60 kg cart moving at 5.0 m/s to the west (W) collides with an 0.80 kg cart moving at 2.0 m/s to the east (E), with a spring constant of 1200 N/m cushioning the collision.

Introduction to Elastic Collisions

During an elastic collision, both momentum and kinetic energy are conserved. The center of mass of the system moves with a constant velocity relative to an external observer, and the internal velocities of the objects change as they interact. This analysis helps us understand the dynamics of such collisions and calculate the outcomes accurately.

Changing to the Center of Mass Frame

One of the most effective methods to analyze this system is by shifting to a moving frame of reference that aligns with the center of mass of the system. Imagine using a video camera moving alongside the center of mass of the two carts. In this frame, the moment of maximum compression of the spring is equivalent to a snapshot taken by a stationary camera at the same instant.

Critical Points in the Collision Process

In this frame, the two carts come to a simultaneous stop when the spring is maximally compressed. At this point, the kinetic energy of the system is zero, as both carts are at rest. This allows us to equate the total initial kinetic energy of the system to the potential energy stored in the compressed spring.

Conservation Laws and Calculations

For a system where there are no external forces, the total momentum is conserved throughout the collision. Similarly, the total kinetic energy is also conserved in an ideal elastic collision. Let's break down the steps to solve the given example:

Calculate the velocity of the center of mass: To find the center of mass velocity, use the formula:

[ V_{cm} frac{{m_1 v_1 m_2 v_2}}{{m_1 m_2}} ]

Where m1 and m2 are the masses of the carts, and v1 and v2 are their respective velocities.

In this case:

[ V_{cm} frac{(0.60)(-5.0) (0.80)(2.0)}{0.60 0.80} -1.7 text{ m/s} ] Transform velocities into the center of mass frame:

The velocities of the carts in the center of mass frame are:

Cart 1: v'1 5.0 1.7 6.7 m/s Cart 2: v'2 2.0 - 1.7 0.3 m/s Set up the conservation of energy equation: The total initial kinetic energy equals the potential energy stored in the spring at maximum compression:

[ frac{1}{2} (m_1 m_2) (V_{cm})^2 frac{1}{2} k x^2 ]

Solving for the compression distance x:

[ x sqrt{frac{(0.60 0.80)(-1.7)^2}{1200}} 0.027 text{ m} ] Calculate the final velocities in the lab frame:

The final velocities in the lab frame are:

Cart 1: v'1' 6.7 - 1.7 5.0 m/s Cart 2: v'2' 0.3 1.7 2.0 m/s

Thus, after the collision, both carts rebound with their original velocities but in the opposite directions.

Conclusion

In summary, elastic collisions are governed by the principles of conservation of momentum and energy. By transforming the system to the center of mass frame, we simplify the calculation and ascertain the outcome of the collision. The example provided illustrates how to apply these principles to solve practical problems involving elastic collisions.