When kinetic energy and momentum are both conserved, the collision is said to be elastic. This indicates that the impact, the system’s total momentum and total kinetic energy are unchanged from before the impact.
The concept of elastic collisions is used to explain a sort of occurrences, including gas particles bouncing off one another and pool ball collisions.
Features of Elastic Collisions:
1.Conservation of Momentum: The system’s overall momentum is conserved in any collision, whether it be inelastic or elastic. The total momentum prior to the collision is equal to the total momentum after the impact, according to the law of conservation of momentum. As long as the system is not impacted by outside factors, this is accurate.
Mathematically, this can be expressed as:
m1u1 + m2u2 = m1v1 + m2v2
where:
m1 and m2 are the masses of the colliding objects,
u1 and u2 are their initial velocities,
v1 and v2 are their velocities after the collision.
2. Conservation of Kinetic Energy: The total kinetic energy of the system stays constant in elastic collisions, in compare to inelastic collisions where some kinetic energy is transformed into other forms of energy (such as heat or sound). Accordingly, the total kinetic energy prior to the impact and the total kinetic energy after the collision are equal.
The equation for kinetic energy conservation is:
1 / 2m1u21 + 1 / 2m2u22 = 1 / 2m1v21 + 1 / 2m2v22
In this equation:
The left side represents the total kinetic energy before the collision,
The right side represents the total kinetic energy after the collision.
Conditions for Elastic Collisions:
Elastic collisions usually happen when there is no deformation or energy loss from heat or friction between the colliding items.
Collisions between hard objects: Two steel balls colliding in an idealised, frictionless environment.
Gas molecules: Gas molecules collide with one another through elastic collisions in which kinetic energy and momentum are both conserved.
Perfect elastic collisions are uncommon in most situations, though, as some energy is naturally lost on deformation, sound, or heat. Elastic collisions are however a helpful idealisation in confined spaces or simplified models.
Types of Elastic Collisions:
Elastic collisions can be classified into two main types:
1.Head-on (One-dimensional) Collisions: When two objects collide head-on, they both go in the same direction, and the collision happens along that path.
The rules of conservation of momentum and kinetic energy dictate how the objects’ velocities vary. Because there are fewer variables concerned, head-on accidents are simpler to mathematically evaluate.
2. Oblique (Two-dimensional) Collisions: The velocity components in both directions must be taken into account when objects collide obliquely, which occurs at an angle. It is necessary to examine momentum conservation in both the horizontal and vertical axes for these kinds of collisions.
Examples of Elastic Collisions:
1.Billiard Balls: Some kinetic energy may be lost in friction or distortion, two pool balls colliding frequently resemble an elastic collision. The balls retain their whole kinetic energy even as they bounce off one another with a transfer of momentum and energy.
2. Gas Molecules: Particles in a gas travel quickly and collide with the container walls and one another in an elastic manner. Because they enable the formulation of important gas laws, such as the ideal gas law, these collisions are important to understand the behaviour of gases.
3. Collisions of Subatomic Particles: Under some circumstances, particles collision between protons or electrons can be roughly described as elastic, particularly in studies where energy conservation is a vital consideration.
Applications:
Elastic collisions are not only a theoretical concept but also have practical applications. For example:
Automobile Safety: Even though these impacts are actually partially inelastic, knowledge of elastic collisions aids in the design of safety systems that absorb energy during a collision, such as airbags and crumple zones.
Sports: Elastic ball-racket or hoop collisions in sports like basketball or tennis can be modeled to forecast the ball’s post-impact trajectory.
Special cases :–
If m1 > > m2 , i.e. target is lighter, then v1 and u1 and v2 = 2u2 – u2
If m1 < < m2 , i.e. target is heavier, then v1 = 2u2 – u1 and v2 = u2
m1 = m2 then v1 = u2 and v2 = u1
If target at rest, i.e., u2 = 0 and m1 > > m2 then v1 = u1 and v2 = 2 u1
Note :-
A simple model of interactions in which kinetic energy and momentum are conserved is called elastic collisions. Forecast the results of particle interactions, and learn more about the behavior of gases by researching elastic collisions.
When kinetic energy and momentum are both conserved, the collision is said to be elastic. This indicates that the system’s total kinetic energy and momentum prior to the impact were equal to those after the collision it.
For a collision to be elastic:
There should be no loss of kinetic energy during the collision.
The momentum of the system must remain conserved.
The colliding objects should not experience permanent deformation or generate heat.
The total momentum before and after an elastic collision remains constant. Mathematically:
m1u1 + m2u2 = m1v1 + m2v2
Here:
m1, m2 are masses,
u1, u2 are initial velocities,
v1, v2 are velocities after the collision.
In elastic collisions, the total kinetic energy of the system before the collision is equal to that of the after the collision. The equation is:
1 / 2m1u21 + 1 / 2m2u22 = 1 / 2m1v21 + 1 / 2m2v22
This means no energy is converted into other forms like heat or sound.
Yes, some examples include:
Billiard Balls: When two billiard balls collide, they often approximate an elastic collision.
Gas Molecules: In ideal gases, collisions between particles are elastic.
Head-on Collisions: The objects collide along the same line of motion, making calculations simpler.
Oblique Collisions: The objects collide at an angle, requiring analysis of momentum conservation in both horizontal and vertical directions.
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