Elastic collision, Definition, Formula, Coefficient of Restitution, Important Topics For JEE Main 2024

Elastic collision : Firstly, definition of collision is to be discussed followed by some real-life examples to deeply understand the concept. Elastic collisions occur when both the momentum and kinetic energy are conserved. Inelastic collisions, on the other hand, happen when only the momentum is conserved but not the kinetic energy. Some of the kinetic energy of the system is transformed into other forms of energy.

Followed by this, we will have a brief discussion on Elastic collision in one dimension. We also have some special cases regarding this collision with different conditions of mass of the bodies colliding. Now elastic collision in two dimensions also needs to be discussed with formula for loss in kinetic energy and momentum conservation. Some worked out examples need to be discussed to get a thorough understanding of the concept.

Brief About Elastic collision :

when two balls collide with each other, and no spring is put between them. At the instant they come into contact, the rear ball has a larger velocity v ₁ and the front ball has a smaller velocity v ₂ . But the surfaces in contact must move equal distance in any time interval as long as they remain in contact. The balls have to be deformed at the contact. The deformed balls push each other and the velocities of the two balls change. The total kinetic energy of the two balls decreases as some energy is converted into the elastic potential energy of the deformed balls. The deformation is maximum (and the kinetic energy minimum) when the two balls attain equal velocities. Total momentum of the balls remains constant. The behaviour of the balls after this depends on the nature of the materials of the balls. If the balls are perfectly elastic, forces may develop inside them so that the balls try to regain their original shapes. In this case, the balls continue to push each other, the velocity of the front ball increases while that of the rear ball decreases and thus the balls separate. After separation, the balls regain their original shapes so that the elastic potential energy is completely converted back into kinetic energy. Thus, although the kinetic energy is not constant, the initial kinetic energy is equal to the final kinetic energy. Such a collision is called an elastic collision.

Elastic collision

Elastic collision : In an elastic collision, the particles regain their shape and size completely after collision.

That is, no fraction of mechanical energy remains stored as deformation potential energy in the bodies. Thus, kinetic energy of a system after collision is equal to kinetic energy of a system before collision. Thus, in addition to the linear momentum, kinetic energy also remains conserved before and after collision.

Head On Elastic Collision

Let the two balls of masses m ₁ and m ₂ collide each other elastically with velocities v ₁ and v ₂ in the directions shown in Fig. Their velocities become v ₁ ′ and v ₂ ′ after the collision along the same line. Applying conservation of linear momentum, we get In an elastic collision kinetic energy before and after collision in also conserved. Hence, Solving Eqs. (iii) and (iv) for and , we get and

Special Case In Elastic Collision :

1. If collision is elastic. i.e., e = 1, then

and

Coefficient of Restitution (e) :

The coefficient of restitution is defined as the ratio of the impulses of recovery and deformation of either body. The most general expression for coefficient of restitution is. Two smooth balls A and B approach each other such that their center’s are moving along the line CD in the absence of external impulsive be u ₁ and u ₂ , respectively, and the velocities of A and B just after collision be v ₁ and v ₂ , respectively. Since, momentum is conserved for the system. ∴ F _ext = 0

Note: e is independent of shape and mass of the object but depends on the material. The coefficient of restitution is constant for two particular objects.

(a) For e = 1

= Impulse of reformation = Impulse of deformation ⇒ Velocity of separation = Velocity of approach ⇒ Kinetic energy is conserved. = Elastic collision

(b) For e = 0

= Impulse of reformation = 0 = Velocity of separation = 0 = Kinetic energy is not conserved = Perfectly inelastic collision

(c) For 0 < e < 1

= Impulse of reformation < Impulse of deformation = Velocity of separation < Velocity of approach = Kinetic energy is not conserved = Inelastic collision

Ex. Two identical balls are approaching towards each other on a straight line with velocity 2 m / s and 4 m / s , respectively. Find the final velocities, after elastic collision between them.

Sol: The two velocities will be exchanged, and the final motion is the reverse of the initial motion for both.