Question

The coefficient of restitution e for a perfectly elastic collision is (A) 1 (B) 0 (C) \(\infty\) (D) \(-1\)

Short answer

For a perfectly elastic collision, the coefficient of restitution is 1. This is because both momentum and kinetic energy are conserved in such a collision. The correct answer is (A) 1.

Step-by-step solution

Understanding the coefficient of restitution

The coefficient of restitution is a measure of the "bounciness" of a collision between two objects. It is defined as the relative speed of separation divided by the relative speed of approach before the collision. Mathematically, it can be represented as follows: \( e = \frac{relative\,speed\,following\,the\,collision}{relative\,speed\,before\,the\,collision} \) For a perfectly elastic collision, we have the conservation of both momentum and kinetic energy.

Conservation of Momentum

Momentum is always conserved in a collision, regardless of whether it is elastic or inelastic. The total momentum before the collision must equal the total momentum after the collision. \( m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} \) Here, \(m_1\) and \(m_2\) are the masses of the two objects, and \(v_{1i}\), \(v_{2i}\), \(v_{1f}\), and \(v_{2f}\) are their initial and final velocities, respectively.

Conservation of Kinetic Energy

In a perfectly elastic collision, kinetic energy is conserved as well. This means that the total kinetic energy before the collision must equal the total kinetic energy after the collision. \( \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \)

Finding the Coefficient of Restitution for a Perfectly Elastic Collision

Using both conservation laws, we can find the value of \(e\) for a perfectly elastic collision. Solving for the relative speeds of separation and approach, we can express \(e\) as follows: \( e=\frac{(v_{1f}-v_{2f})}{(v_{2i}-v_{1i})} \) After solving the equations for conservation of momentum and conservation of kinetic energy simultaneously and simplifying, we obtain: \( e = 1 \) Thus, for a perfectly elastic collision, the coefficient of restitution is 1.

Answer

The correct answer is (A) 1, which is the coefficient of restitution for a perfectly elastic collision.

Key concepts

Elastic Collision

An elastic collision occurs when two objects collide and then bounce off each other without any loss of kinetic energy in the system. These collisions are often thought of as "ideal" because the total kinetic energy and the total momentum remain constant before and after the event. In the real world, perfect elastic collisions rarely happen, as most collisions involve some energy loss due to factors like sound or heat.
In an elastic collision, both the conservation of momentum and conservation of kinetic energy principles apply:

• The total momentum before the collision equals the total momentum after the collision.
• The total kinetic energy before the collision is the same as after the collision.

These attributes make elastic collisions a critical concept for understanding mechanical interactions between objects.

Momentum Conservation

Momentum conservation is a fundamental concept in physics that states the total momentum of a closed system remains constant before and after a collision, provided no external forces are acting on it. Momentum \( p \) is the product of mass and velocity, \( p = mv \).
For two colliding objects, this can be expressed as:

• Initial momentum of object one: \( m_1v_{1i} \)
• Initial momentum of object two: \( m_2v_{2i} \)
• Final momentum of object one: \( m_1v_{1f} \)
• Final momentum of object two: \( m_2v_{2f} \)

The conservation equation is:\( m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} \)
This equation is crucial for analyzing collisions and interactions in systems where external forces can be ignored.

Kinetic Energy Conservation

Another key principle in elastic collisions is the conservation of kinetic energy. Unlike inelastic collisions, where some kinetic energy is transformed into other forms like thermal energy, elastic collisions maintain the total kinetic energy.
The kinetic energy of an object is given by the formula:\( KE = \frac{1}{2}mv^2 \)This takes into account both mass and the square of velocity.
For an elastic collision between two objects, the equation becomes:\( \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \)
This conservation helps verify that a collision is truly elastic and provides vital insights into the nature of the interaction between the colliding bodies.

Velocity

Velocity, defined as speed in a given direction, is a vector quantity crucial in analyzing motion and collisions. In the context of elastic collisions, understanding changes in velocity allows us to apply the conservation laws effectively.
The relative velocity before and after the collision is particularly important in determining the coefficient of restitution \( e \). This coefficient helps identify the degree of elasticity of a collision:

• If \( e = 1 \), the collision is perfectly elastic.
• If \( e = 0 \), it's a perfectly inelastic collision.
• For any other values between 0 and 1, the collision is partially elastic.

Calculating velocities before and after collisions aids in understanding the dynamics and impact of the collision events.

Physics Education

Physics education provides foundational understanding of principles governing the universe, like momentum and energy conservation seen in elastic collisions. These concepts are not just academic exercises; they are applicable in various real-world scenarios, from sports to engineering designs.
Studying collisions helps students grasp how objects interact under different conditions, fostering an intuitive feel for dynamics and mechanics. This deeper understanding equips learners to analyze situations logically and enhances their problem-solving skills.

• Engagement with practical experiments helps reinforce theoretical knowledge.
• Using simulations and visualizations can make abstract concepts more tangible.

Ultimately, physics education inspires curiosity and innovation, paving the way for future scientific advancements.