Collisions - Coefficient of Restitution e

When two objects collide, they behave differently depending on how much energy they lose. The coefficient of restitution (e) is a number that tells us how bouncy or sticky a collision is. It measures the relative speed at which two objects move away from each other compared to how fast they were moving toward each other before hitting. This helps us understand if a collision is elastic (objects bounce apart), inelastic (objects lose some energy), or perfectly inelastic (objects stick together). Understanding this concept is important for solving collision problems in physics.

The negative of the ratio of the relative velocity of separation to the relative velocity of approach.

Mathematically: e = -\[\frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}\]

• Head-on collision: Colliding objects move along the same straight line before and after the collision. Vector treatment is not necessary; only algebraic values of velocities are used.
• Range of coefficient of restitution: For any collision, 0 ≤ e ≤ 1
• Perfectly elastic collision: e = 1 (objects bounce apart completely; no energy is lost)
• Perfectly inelastic collision: e = 0 (objects stick together after collision; maximum energy is lost)
• Partially elastic collision: 0 < e < 1 (objects bounce apart but lose some energy)

Step-by-Step Explanation:

Understanding Relative Velocities

Before two objects collide, they move toward each other with different speeds. The relative velocity of approach tells us how fast they are getting closer:

• u_a = u₂ − u₁

Collision Occurs

During the collision, forces act on both objects, changing their velocities.

Objects Move After Collision

After a collision, the objects either move in the same direction or opposite directions. The relative velocity of separation tells us how fast they are moving apart:

• v_s = v₂ − v₁

Calculate Coefficient of Restitution

The coefficient of restitution compares the separation speed to the approach speed:

e =\[-\frac{v_{s}}{u_{a}}=\frac{v_{1}-v_{2}}{u_{2}-u_{1}}\]

For Elastic Collisions (e = 1):

In an elastic collision, two conservation laws apply:

Conservation of Momentum:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Conservation of Kinetic Energy:
\[\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\]

When we solve these equations together, we get:
u₂ − u₁ = v₁ − v₂

This means: e = 1

For Perfectly Inelastic Collisions (e = 0):

In this case, the two objects stick together after collision:

• v₁ = v₂ (or both velocities are zero)

Therefore: e = \[\frac{v_1-v_2}{u_2-u_1}=\frac{0}{u_2-u_1}\] = 0

• Classifies collisions: Tells us whether a collision is elastic, inelastic, or perfectly inelastic.
• Predicts object behavior: Helps determine if objects will bounce apart or stick together after collision.
• Real-world applications: Used to analyze collisions in sports (ball bouncing), accidents (car crashes), and engineering (impact testing).
• Energy analysis: Shows how much kinetic energy is retained or lost during a collision.
• Problem-solving: Essential for solving collision problems using physics principles.

Elastic Collision (e ≈ 1):

• A billiard ball hitting another billiard ball on a pool table. Both balls bounce apart, and very little energy is lost.
• Two steel balls are colliding in a Newton's cradle (the swinging ball toy).

Inelastic Collision (0 < e < 1):

• A tennis ball hits a wall and bounces back at a lower speed than it arrived.
• A rubber ball dropped from a height, bouncing multiple times until it stops.

Perfectly Inelastic Collision (e = 0):

• A car crash where two vehicles collide and stick together after the collision.
• A bullet hits and embeds itself in a wooden block.
• Two pieces of clay are sticking together after colliding.