Introduction to Collisions
When we think about collision, a simple picture of two objects striking head to head comes into our mind. For example, two cars meeting head to head will result in an accident. Similarly, a person hitting a football, or the carrom pug hitting each other in a carromboard. These events are everyday examples of collisions which may look easier but in physics the word ‘Collision’ has a substantial meaning and is studied vastly in mechanics and classical mechanics. Collision doesn’t only refer to a crash or a damage, but it stands for the physical contact of two or more than two bodies. It also deals with the force that makes them collide, which is sudden.
To understand about forces, momentum, energies and their nature at various conditions, collision and its types (Elastic and inelastic collisions) are studied thoroughly in physics and engineering. They also carry the law of conservation of energy and momentum. The total momentum of the colliding systems always remains conserved when no external force acts on it. However, the kinetic energy may or may not remain conserved — and this difference helps us categorize collisions into two main types: elastic collisions and inelastic collisions. It applies from two dimensional to three dimensional objects and hence, it is necessary to figure out the characteristics of higher and massive bodies by assuming smaller bodies. Below is given the brief.
Definition of Elastic and Inelastic Collisions
Basically, in collisions we observe two quantities viz kinetic energy and the momentum of the involved bodies. When two bodies in motion (or either one at rest) collide, they acquire momentum. The resulting total momentum of the system combining both objects is always conserved. However, the kinetic energy may differ before and after the collision.
Elastic Collision
When the objects are head on, the both quantities: momentum and kinetic energy are conserved. After the collision, the objects move away with their same initial momentum and the kinetic energy. Hence, no loss of total kinetic energy occurs and no energy is transformed into sound, heat or any other form.
In practice, elastic collisions are hard to attain. Very rare cases like the collision of gas molecules and the striking of billiard balls come closer to elastic collisions.
Inelastic Collision
An inelastic collision only conserves the momentum after collisin while kinetic energy is disturbed. The kinetic energy is lost in the form heat, sound or deformation occurred during collision.
When a perfect inelastic collision happens, two colliding bodies merge together and after the collision they move as a combined mass. It is the case where the maximum loss of kinetic energy takes place. However, the momentum remains the same..
Conservation Laws in Collisions
Since only two quantities ‘Kinetic Energy and Momentum’ come into play for the collision, the conservation laws also look for the same, in case of collision.
- Law of Conservation of Momentum
The law of conservation of momentum states that, the total momentum before and after the collision is equal. It means that two colliding bodies will move with the same momentum, before and after the collision . If two bodies of masses m1​ and m2​ and velocities u1​ and u2 respectively collide then after collision their velocity will be v1 and v2 following the comnservation equation:
m1​u1​+m2​u2​=m1​v1​+m2​v2​ [Equation 1]
This law and the equation for momentum is valid for both types of collisions.
- Law of Conservation of Kinetic Energy
The law of conservation of kinetic energy is valid only for the condition of elastic collisions. It says that, the total kinetic energy of a system remains constant during an elastic collision.
So, for the elastic collision the kinetic energy follows the equation:
1/2​ m1​u1^2​+1/2​ m2​u2^2​=1/2​ m1​v1^2​+1/2​ m2​v2^2​ [Equation 2]
*Point to be noted: For inelastic collisions, some energy is wasted into other forms.
Elastic Collisions
An elastic collision is one which conserves both the quantities kinetic energy and momentum. Thus, a 100% elastic collision is difficult to achieve. Although if the masses are minute or of microscale, the collision is likely to be elastic.
These types of collisions are common some systems like, collision of gas molecules,
elastic scattering of electrons or the collisions of steel balls or billiard balls.
Properties of Elastic Collisions
- Total kinetic energy is conserved before and after the collision.
- Total momentum is conserved in all directions.
- The velocities can be exchanged but they can never stick together.
- Other factors like internal energy, temperature, and deformation remain unchanged.
- The impact period is very short.
- From conservation of momentum:
If we have two bodies colliding in a straight line (one-dimension) with masses m1​ and m2 colliding with initial velocities u1​ and u2​ then, their final velocities after elastic collision will be v1​ and v2​.
m1​u1​+m2​u2​=m1​v1​+m2​v2​
From conservation of kinetic energy:
1/2​ m1​u1^2​+1/2​ m2​u2^2​=1/2​ m1​v1^2​+1/2​ m2​v2^2​
By solving these two equations, we get:
v1 = (m1−m2)u1+2m2u2 / (m1+m2) [Equation 3]
v2 ​= ​(m2​−m1​)u2​+2m1​u1​​/ (m1+m2) [Equation 4]
This gives the calculation of velocities after an elastic collision.
Inelastic Collisions
In an inelastic collision, only momentum is conserved while kinetic energy is lost.This energy is transformed to other forms such as sound, heat, or deformation energy. Objects with heavier masses generally experience this type of collision.
Properties of Inelastic Collisions
- Momentum remains constant.
- Kinetic energy is lost into other forms of energy.
- The colliding bodies may get their shape or size deformed permanently.
- There is a loss of energy due to heat or sound during the collision.
- The objects can permanently stick together and move with different kinetic energy (perfectly inelastic case).
- Most of the real-world collisions are inelastic.
Some examples of inelastic collisions may be a car striking on a wall or another vehicle, a gun shot with the bullet embedding into an object, a clay ball hitting a surface etc.
Perfectly Inelastic Collisions and Coefficient of Restitution
Perfectly Inelastic Collisions
When two bodies collide and stick together, finally moving as a single body, they give the maximum condition of collision making it perfectly inelastic. As described above, now they have different kinetic energy then the initial condition. For two masses m1 and m2, with initial velocities u1 and u2, the final velocity of both combinedly becomes,
v = m1​u1​+m2​u2 / (m1+m2)​​ [Equation 5]
Coefficient of Restitution (e)
The coefficient of restitution is calculated as the ratio of relative velocity after collision to the relative velocity before collision.
e = Relative velocity after collision / Relative velocity before collision
= v2−v1/u1−u2
- For perfectly elastic collisions, e=1.
- For perfectly inelastic collisions, e=0.
- For partially inelastic collisions, 0<e<1.
This helps to describe the nature of energy after the collision occurs.
Mathematical Formulation of Elastic and Inelastic Collisions
For simplicity, we study the collision mathematically in one-dimension.
Elastic Collision Equations
For elastic collision we can use the law of conservation as equation 1 and equation2:
m1​u1​+m2​u2​=m1​v1​+m2​v2​
1/2​ m1​u1^2​+1/2​ m2​u2^2​=1/2​ m1​v1^2​+1/2​ m2​v2^2​
Solving these equation get,
v1 = (m1−m2)u1+2m2u2 / (m1+m2)
v2 ​= ​(m2​−m1​)u2​+2m1​u1​​/ (m1+m2)
Thus we can predict the final velocities of those colliding objects using above equations for v1 and v2.
Inelastic Collision Equations
In an inelastic collision, the momentum equation is the same as equation 1:
m1​u1​+m2​u2​=m1​v1​+m2​v2​
For a perfectly inelastic collision, the final combined velocity of bothobjects is:
v = m1​u1​+m2​u2 / (m1+m2)​​
The loss of kinetic energy in this case is obtained as:
ΔKE = 1/2 ​m1​m2 / m1​+m2 ​​(u1​−u2​)^2 [Equation 6]
This is the energy that is lost in an inelastic collision in the form of sound, heat, or deformation.
Examples of Elastic and Inelastic Collisions in Daily Life
Elastic Collision
- Billiard Balls: When a player strikes one billiard ball with another on a surface, energy and momentum remains nearly constant.This is a closer example of an elastic collision. In practice, a little heat or sound is lost.
- Atoms and Molecules: At the molecular level, the collisions are approximately elastic.
- Newton’s Cradle: Newton’s cradle is a toy which has metal balls swinging freely. When they collide they form a nearly elastic collision.
Inelastic Collision
- Car Crashes: The metal and also the striking surface gets deformed producing sound energy during a crash. This converts kinetic energy into heat, sound and deformation.
- Clay Ball on Floor: When clay hits the ground it gets stuck. Hence, all the kinetic energy is transformed. Thus, we get a perfectly inelastic collision.
- Football Kick: When a player hits a football, the ball deforms first and regains its shape later. Hence, kinetic energy is transformed into deformation. Inelastic collision is found in this case.
Experimental Verification and Graphical Analysis
Experimental Verification
Experiments can clearly demonstrate the conservation laws in collisions. In physics and engineering labs, a linear air trajectory almost without any friction is used to study how gliders can crash into each other.
Procedure:
- Two gliders of different masses are placed on the air track.
- They are brought into motion toward each other.
- Their velocities before and after collision are measured with the help of sensors or photogates.
- In observation it is obtained that,
- Total momentum before ≈ total momentum after (momentum conserved).
- Kinetic energy before ≈ kinetic energy after (only for elastic cases).
- Total momentum before ≈ total momentum after (momentum conserved).
This shows the momentum conservation in both types of collisions and energy conservation only in elastic cases.
Graphical Analysis
Graphical analysis of a collision is necessary to visualize the collision data and interpret it correctly.
- Momentum vs. Time Graph: It shows a constant total momentum of the system before and after collision.
- Kinetic Energy vs. Time Graph: It displays the nature of collisions (either elastic or inelastic) and interprets the result with time.
- Velocity-Time Graph: This helps to compare the changes in the velocities of participating objects individually, before and after collision.
Applications and Significance in Science and Engineering
Collisions can be natural or can be obtained in a virtual environment for various purposes. Some of the applications are given below:
- Automobile Safety Design
Car companies study inelastic collisions to predict the loss in the kinetic energy and hence make crumple zones. This reduces the physical loss in health and wealth during crashes. These zones convert kinetic energy into deformation energy safely and reduce the probable losses.
- Sports Science
In games like cricket, baseball, or football, understanding the physics of collision is necessary. It helps to improve our games and also the equipment design. For example, balls of elastic or inelastic nature can be designed.
- Particle Physics
Elastic and inelastic collisions in the subatomic levels give us important information about their structure. The fundamental forces can also be studied with it. High-energy devices like the Large Hadron Collider work on the basis of collision.
- Engineering Structures
Engineers study the impact behavior while designing buildings, bridges etc. and make them able to withstand collisions and impacts safely.
- Astronomy and Space Science
Collisions can also occur in the space, betwwen the heavenly bodies. Thus, scientists study their collision to study the formation of celestial objects and crater impacts.
- Material Testing
The elasticity, strength and durability of a material can be studied by analyzing its nature under collision.
- Robotics and Automation
Artificial collisions are studied in robots to make grippers, sensors, and parts that can handle simple natural collisions. They carefully simulate how things work in nature.
Conclusion
Collision is a physical action that occurs frequently in nature. It can also be created in an artificial manner. Whether the collision is of microscopic bodies or macroscopic ones, they follow the conservation laws for momentum. However, energy conservation may come up with different cases. Elastic and inelastic collisions vastly differ in energy. A perfectly elastic collision conserves whole kinetic energy while a perfectly inelastic collision transforms all its energy and both colliding objects are merged together and move as a one body. But, both cases are not very common. The lost energy in an inelastic collision shows up as heat, sound or deformation energy.
Collision incidents specially, the inelastic one is common in real situations. Some incidents have greater impacts and are harmful too. Thus, both types are studied well by scientists and safety methods are built up. Scientists and engineers design safer vehicles, better materials, and efficient machines by studying the collision effects on a material. Thus, whether it’s a gas molecule collision or a car crash, it is an important topic for physics and other technological sectors.
References
https://www.geeksforgeeks.org/physics/difference-between-elastic-and-inelastic-collision/
https://en.wikipedia.org/wiki/Elastic_collision
https://en.wikipedia.org/wiki/Inelastic_collision
https://www.vedantu.com/physics/elastic-and-inelastic-collisions
Johnson, R. E. (2012). Introduction to atomic and molecular collisions. Springer Science & Business Media.
Khare, S. P. (2001). Basics of Collisions. In Introduction to the Theory of Collisions of Electrons with Atoms and Molecules (pp. 1-14). Boston, MA: Springer US.
