OCR Specification focus:
‘Distinguish perfectly elastic and inelastic collisions; discuss kinetic energy changes and real-world contexts.’
Collisions are central to understanding how momentum and energy behave during interactions between objects, allowing physicists to predict outcomes in both theoretical and real-world systems.
Elastic and Inelastic Collisions
In all collisions, the principle of conservation of momentum applies. The total momentum of a system before and after a collision remains constant, provided no external forces act on it. However, the way kinetic energy behaves distinguishes elastic and inelastic collisions.
Conservation of Momentum
EQUATION
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Conservation of Momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
m = mass of the object (kg)
u = velocity before collision (m s⁻¹)
v = velocity after collision (m s⁻¹)
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Momentum, a vector quantity, means direction must always be accounted for when applying the conservation law. This applies in both one-dimensional and two-dimensional collisions.
Kinetic Energy in Collisions
While momentum is always conserved, kinetic energy is not necessarily conserved during a collision. Some energy may be transformed into other forms, such as sound, heat, or deformation energy.
EQUATION
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Kinetic Energy (Eₖ) = ½mv²
Eₖ = kinetic energy (J)
m = mass (kg)
v = velocity (m s⁻¹)
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By comparing the total kinetic energy before and after impact, collisions can be classified as elastic or inelastic.
Perfectly Elastic Collisions
Definition and Characteristics
Perfectly Elastic Collision: A collision in which both momentum and kinetic energy are conserved.
In a perfectly elastic collision, no kinetic energy is converted to other forms.
One-dimensional elastic collision: the bodies separate with kinetic energy conserved and momentum unchanged for the system. Vector labels make the before-and-after velocities explicit. This diagram aligns with the idealised case discussed in kinetic theory and simple cart collisions. Source
The total kinetic energy of the system before and after the collision is identical. This idealised condition is extremely rare in macroscopic systems but can be approximated in microscopic interactions, such as between gas molecules.
In such collisions:
Total momentum before = total momentum after.
Total kinetic energy before = total kinetic energy after.
There is no permanent deformation or loss of energy to heat or sound.
The objects rebound without energy dissipation.
For example, atomic collisions in gases are often modelled as perfectly elastic because the internal energy states of atoms remain unchanged during these brief interactions.
Mathematical Treatment
To test if a collision is perfectly elastic, one compares the total kinetic energies before and after impact. If they are identical, the collision is elastic. This can be expressed as:
EQUATION
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Kinetic Energy Conservation: ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²
m = mass (kg)
u = velocity before (m s⁻¹)
v = velocity after (m s⁻¹)
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Elastic collisions also obey the relative speed condition, where the speed of approach before collision equals the speed of separation after collision.
Real-World Contexts of Elastic Collisions
While perfectly elastic collisions are an idealisation, several physical systems closely approximate them:
Gas molecule collisions, which underpin the kinetic theory of gases.
Collisions between steel ball bearings or Newton’s cradle devices.
Subatomic particle interactions studied in accelerators, where energy losses are negligible.
These examples demonstrate how energy conservation principles are applied to understand microscopic and macroscopic dynamics.
Newton’s cradle demonstrates near-elastic momentum transfer between steel spheres with small energy losses to sound and heat. It visualises equal and opposite impulses and near-conservation of kinetic energy in short, central impacts. Note: as discussed in the notes, real devices are not perfectly elastic due to dissipative effects. Source
Inelastic Collisions
Definition and Characteristics
Inelastic Collision: A collision in which momentum is conserved but kinetic energy is not; some kinetic energy is transformed into other energy forms.
In inelastic collisions, although total momentum remains constant, part of the kinetic energy is converted into internal energy, heat, sound, or deformation. This energy transformation means the objects typically do not rebound with their initial speed.
Key characteristics include:
Momentum conservation still holds for the system.
Kinetic energy decreases after the collision.
Energy is dissipated as heat, sound, or permanent deformation.
The objects may stick together or move apart more slowly.
When objects stick together after impact, the event is classified as a perfectly inelastic collision.
Perfectly inelastic collision: two equal masses approach with equal and opposite speeds, stick on impact, and come to rest. Momentum is conserved, but kinetic energy decreases, converted to internal energy, sound, and deformation. The clean “before–after” layout mirrors the equations given for the common final velocity. Source
Perfectly Inelastic Collisions
Perfectly Inelastic Collision: A collision in which the colliding objects stick together after impact, moving as a single combined mass.
In perfectly inelastic collisions, the loss of kinetic energy is maximised for a given system, though momentum remains conserved. These are common in real-world scenarios such as car crashes or clay blobs colliding and joining.
EQUATION
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Momentum in Perfectly Inelastic Collision: m₁u₁ + m₂u₂ = (m₁ + m₂)v
m = mass (kg)
u = velocity before (m s⁻¹)
v = common velocity after (m s⁻¹)
—-----------------------------------------------------------------
This equation demonstrates that after impact, both bodies share the same final velocity, determined by their combined momentum and total mass.
Energy Transformations in Inelastic Collisions
In all inelastic collisions, some of the system’s mechanical energy is converted into non-mechanical forms. The reduction in kinetic energy corresponds to work done on deformation, friction, or sound production.
Common transformations include:
Thermal energy, due to friction and internal vibrations.
Sound energy, generated by the impact.
Elastic potential energy, temporarily stored in compressed or deformed regions.
Permanent deformation energy, resulting in structural changes.
Real-World Contexts of Inelastic Collisions
Examples include:
Vehicle collisions, where kinetic energy becomes heat, sound, and body deformation.
Sports impacts, such as a cricket ball striking a bat, where energy is lost as sound and deformation.
Meteorite impacts, converting enormous kinetic energy into heat and seismic energy.
These situations illustrate that while kinetic energy is not conserved, the total energy of the system remains constant in accordance with the principle of conservation of energy.
Comparing Elastic and Inelastic Collisions
Understanding the distinction between elastic and inelastic collisions allows physicists to model a wide range of physical interactions accurately.
Elastic collisions conserve both momentum and kinetic energy.
Inelastic collisions conserve momentum only, with kinetic energy partially transformed.
The degree of elasticity depends on material properties and collision dynamics.
This comparison reinforces how momentum conservation remains a universal principle, while kinetic energy conservation depends on the nature of the collision and the system’s internal processes.
FAQ
The degree of elasticity depends on how much kinetic energy is converted into other forms during the collision.
Material properties play a major role. Hard, non-deformable materials like steel lose little energy, while soft materials such as rubber or clay dissipate more.
Surface texture and temperature also affect elasticity; rough or warm surfaces tend to absorb more energy.
The coefficient of restitution quantifies elasticity on a scale from 0 (perfectly inelastic) to 1 (perfectly elastic).
Momentum conservation arises from Newton’s third law of motion — equal and opposite forces act between colliding objects over the same time interval.
These internal forces cause equal and opposite momentum changes that cancel when summed across the system, provided no external forces act.
Kinetic energy, however, depends on speed squared. If any energy is converted into heat, sound, or deformation, total kinetic energy decreases while total momentum remains constant.
Energy loss is determined by comparing total kinetic energy before and after a collision.
Typical method:
Use motion sensors or light gates to measure velocities of objects before and after impact.
Calculate kinetic energies using ½mv².
The difference represents energy transformed into other forms.
For rolling or sliding objects, frictional effects must be minimised or corrected to isolate energy lost solely through the collision itself.
No. Whether a collision is elastic or inelastic is independent of reference frame.
Although measured velocities differ between observers, the total kinetic energy and momentum calculated for the system transform consistently between frames.
If kinetic energy is conserved in one inertial frame, it will be conserved in all; the same applies for non-conservation. This invariance ensures that the classification of a collision as elastic or inelastic is absolute, not relative.
Gas molecule collisions are nearly elastic because intermolecular forces act only during the brief collision and do not cause lasting deformation or heating.
The molecules are extremely small and rigid, with negligible internal structure to absorb energy.
Energy exchanges are purely between translational kinetic energies, not internal vibrations or rotations.
In ideal gas models, collisions are assumed perfectly elastic to simplify calculations of pressure, temperature, and molecular speed distributions.
Real gases deviate slightly due to weak attractive forces, especially at high pressures or low temperatures.
Practice Questions
Question 1 (2 marks)
A moving ball collides elastically with another identical stationary ball on a smooth horizontal surface.
State two physical quantities that are conserved in this collision and explain briefly what this means.
Mark Scheme:
1 mark for identifying momentum as conserved.
1 mark for identifying kinetic energy as conserved.
(Explanation that total momentum/total kinetic energy before = total momentum/total kinetic energy after the collision is implied and not separately credited.)
Question 2 (5 marks)
A trolley of mass 1.5 kg moving at 2.0 m s⁻¹ collides head-on and sticks to a stationary trolley of mass 1.0 kg on a frictionless track. The collision is perfectly inelastic.
(a) Calculate the velocity of the combined trolleys after the collision. (2 marks)
(b) Calculate the total kinetic energy before and after the collision, and determine the percentage loss of kinetic energy. (3 marks)
Mark Scheme:
(a)
1 mark for correct application of momentum conservation:
(1.5 × 2.0) + (1.0 × 0) = (1.5 + 1.0) v
1 mark for correct final velocity: v = 1.2 m s⁻¹
(b)
1 mark for correct initial kinetic energy:
KE_before = ½ × 1.5 × (2.0)² = 3.0 J
1 mark for correct final kinetic energy:
KE_after = ½ × (2.5) × (1.2)² = 1.8 J
1 mark for correct percentage loss:
(3.0 − 1.8) / 3.0 × 100 = 40% loss
(Allow rounding to one or two significant figures. Units required for full marks in calculation steps.)
