Collisions in One Dimension
Analysis of elastic and inelastic collisions in a straight line, applying conservation laws.
What is Collisions in One Dimension?
An event in which two or more bodies exert forces on each other for a short time.
Key formula / rule: Conservation of Linear Momentum
Key points
- Define and differentiate between elastic and inelastic collisions.
- Apply the principle of conservation of linear momentum to solve collision problems.
- Apply the principle of conservation of kinetic energy to elastic collision problems.
- Calculate final velocities of objects after one-dimensional collisions.
Common exam trap
Applying conservation of kinetic energy to inelastic collisions.
Definitions
- Term
Collision
- Meaning
An event in which two or more bodies exert forces on each other for a short time.
- Term
Elastic Collision
- Meaning
A collision in which both linear momentum and kinetic energy are conserved.
- Term
Inelastic Collision
- Meaning
A collision in which linear momentum is conserved, but kinetic energy is not.
- Term
Perfectly Inelastic Collision
- Meaning
A type of inelastic collision where the colliding bodies stick together after impact and move with a common velocity.
- Term
Coefficient of Restitution (e)
- Meaning
A dimensionless quantity that describes the elasticity of a collision, defined as the ratio of the relative speed of separation to the relative speed of approach.
Learning objectives
Define and differentiate between elastic and inelastic collisions.
Apply the principle of conservation of linear momentum to solve collision problems.
Apply the principle of conservation of kinetic energy to elastic collision problems.
Calculate final velocities of objects after one-dimensional collisions.
Understand the concept of the coefficient of restitution.
Formulae
- Name
Conservation of Linear Momentum
- Note
Applies to all types of collisions in the absence of external forces. u represents initial velocity, v represents final velocity.
- Expression
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Name
Conservation of Kinetic Energy (Elastic Collisions)
- Note
Only applicable for perfectly elastic collisions.
- Expression
½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²
- Name
Coefficient of Restitution (e)
- Note
Relates relative velocities before and after collision. e=1 for elastic, 0 < e < 1 for inelastic, e=0 for perfectly inelastic.
- Expression
e = (v₂ - v₁) / (u₁ - u₂)
- Name
Velocity after collision (special case: m₁ collides with stationary m₂)
- Note
General formulae for 1D collisions. For stationary m2, u2=0. For elastic collision, e=1. For perfectly inelastic, e=0 and v1=v2.
- Expression
v₁ = ((m₁ - em₂)/(m₁ + m₂))u₁ + ((1+e)m₂/(m₁ + m₂))u₂ v₂ = ((m₂ - em₁)/(m₁ + m₂))u₂ + ((1+e)m₁/(m₁ + m₂))u₁
Prerequisites
Linear Momentum
Newton's Laws of Motion
Kinetic Energy
Work and Energy
Vectors
Common mistakes
Applying conservation of kinetic energy to inelastic collisions.
Forgetting to treat momentum as a vector quantity (though in 1D, direction is handled by signs).
Incorrectly calculating relative velocities after collision.
Assuming all collisions are elastic.
Not considering the system of both colliding bodies.
Keywords
Collision
One Dimension
Elastic Collision
Inelastic Collision
Perfectly Inelastic Collision
Conservation of Momentum
Conservation of Kinetic Energy
Coefficient of Restitution
Impulse
Practice preview
A body of mass 4 kg moving at 10 m/s collides with a stationary body of mass 6 kg. If the collision is elastic, what is the speed of the first body after collision?…
medium
A body of mass m1 moving with velocity v1 collides elastically with a stationary body of mass m2. If m1 = m2, what is the velocity of m1 after collision?…
easy
A 10 g bullet moving at 500 m/s strikes and embeds itself into a stationary block of wood of mass 1 kg. What is the velocity of the block immediately after the bullet embeds?…
medium