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Elastic Collision

The concept of collision is described by the use of conservation of momentum and momentum. The conservation of momentum is based on the conservation of energy and kinetic energy concept. The collision takes place between two or more objects. When objects collide with each other than each object feels small amount of force for a small period of time. This force alters the momentum of the colliding objects.

But in case of isolated system of particles, the momentum is in conservative state. So the momentum of individual particle is changed but total momentum of system remains constant. The detection procedure of collision is only depending on the type of collision. There are two types of collision that can be is elastic or inelastic. Both are different in their properties like the Kinetic energy conservation is done in case of elastic collisions while it is converted into other forms of energy in case of inelastic collision but there is conservation of momentum in both the cases. Here we are discussing all about the elastic collision in one and two dimension, its mathematical formula and their comparison.

Elastic Collision

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Elastic Collision Definition

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A type of collision in which 2 Laws are followed viz.
  1. The Law of Conservation of Momentum and
  2. The Law of Conservation of Kinetic Energy
A type of collision in which there is no net conversion of kinetic energy in any other form before or after the collision is called Elastic Collision.If we consider the collision between two small objects then during the collision the kinetic energy first gets converted into potential energy and finally this potential energy again gets converted into kinetic energy. In this way the kinetic energy is conserved.

Perfectly Elastic Collision

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Perfectly Elastic collision will be the ones in which kinetic energy is fully conserved, that is, it is not at all lost or converted into any other form.
In case of small molecules or atoms (individual atoms), this is not at all possible, the collisions are not at all perfectly elastic. This is because during the collision, the exchanging of kinetic energy takes place between the translational motion of molecules and the internal degrees of freedom.

Perfectly elastic collisions are an ideal condition collision and hence are never fully realized even between macroscopic objects.
Example: The collision between two billiard balls can be regarded as perfectly elastic collision approximately.

No energy is being radiated away (kinetic energy) or consumed internally.

Example : Bouncing of a ball when it hits the surface.

    Perfect Elastic Collision


    Elastic Collision Formula

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    One dimensional Newtonian equation for elastic collision:
    Let two objects with mass m1 and m2.
    Initial velocity of object 1 = u1,
    Initial velocity of object 2 = u2
    Let v1 and v2 be the final velocities respectively.

    Applying conservation of momentum principle, we get:

    m1 u1 + m2 u2 = m1 v1 + m2 v2 .....................(1)

    Also applying principle of kinetic energy conservation:

    $\frac{1}{2}$ m1 u12 + $\frac{1}{2}$ m2 u22 = $\frac{1}{2}$ m1 v12+ $\frac{1}{2}$ m2 v22 .. . . . (2)

    Solving the above equations for v1 and v2 we get:
    v1 = $\frac{u_{1} (m_{1} - m_{2})+ 2m_{2} u_{2}}{m_{1} + m_{2}}$.....................(3)

    Similarly
    v2 = $\frac{u_{2} (m_{2} - m_{1})+ 2m_{1} u_{1}}{m_{1} + m_{2}}$.....................(4)

    Also considering the trivial case such that there has been no collision yet:
    v1 = u1 and v2 = u2................................(5)

    yet another property : v1 – v2 = u2 – u1.

    Elastic Collision Momentum

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    If we consider special relativity then for momentum: P = $\frac{m v}{\sqrt{[1 – \frac{v^{2}}{c^{2}}]}}$Here,
    P = momentum of a massive particle.
    V =velocity of the particle
    C = speed of light

    Considering the center of momentum frame ( total momentum = 0 )
    p1 = - p2
    p12 = p22
    $\sqrt{(m_{1}^{2} c^{4} + p_{1}^{2} c^{2})}$ + $\sqrt{( m_{2}^{2} c^{4} + p_{2}^{2} c^{2} )}$ = E
    p1 = $\sqrt{( E_{4} – 2 E_{2} m_{1}^{2} c^{4} – 2 E^{2} m_{2}^{2} c^{4} + m_{1}^{4}c^{8} – 2 m_{1}^{2} m_{2}^{2} c^{8} + m_{2}^{4} c^{8})}$u1 = - v1
    Here m1 and m2 are the rest masses of particle 1 and 2
    u1 and u2 : Initial velocity
    v1 and v2 : Final velocity
    p1: Momentum of first colliding particle
    p2: Momentum of second colliding particle.

    Conservation of Momentum Elastic Collision

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    In an Elastic collision the momentum is conserved or the law of conservation of momentum before and after the collision is same.
    Hence for an elastic collision:
    m1 u1 + m2 u2 = m1 v1 + m2 v2 . . . . . . . .(1)Here,
    Initial velocity of object 1 = u1,
    Initial velocity of object 2 = u2,
    Final Velocity of object 1 = v1,
    Final Velocity of object 2 = v2.
    Using this and assuming that m1 >> m2
    v1 = u1
    v2 = u2

    Elastic Collision Examples

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    Various examples of elastic collision are:
    Elastic Collision Examples
    1. When we throw a ball on the floor, it bounces back. This is an example of elastic collision where both momentum and kinetic energy are conserved.
    2. The collision between the atoms is also an example of elastic collision.
    3. The collision between two billiard balls is an example of elastic collision.

    Elastic Collision in One Dimension

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    For one dimension elastic collision applying conservation laws for two colliding objects with velocities of u1 and u2 and final velocities as v1 and v2:
    Applying conservation of momentum principle we get:
    m1 u1 + m2 u2 = m1 v1 + m2 v2 ......................................(a)

    Also applying principle of kinetic energy conservation:
    $\frac{1}{2}$ m1 u12 + $\frac{1}{2}$ m2 u22 = $\frac{1}{2}$ m1 v12 + $\frac{1}{2}$ m2 v22 ...........................................(b)

    Solving the above equations for v1 and v2 we get:
    v1 = $\frac{u_{1}(m_{1} - m_{2})}{m_{1} + m_{2}}$.
    v2 = $\frac{u_{2}(m_{2} - m_{1}) + 2 m_{1} u_{1})}{m_{1} +m_{2}}$ ...........(c)


    Also considering the trivial case such that there has been no collision yet:
    v1 = u1 and v2 = u2.

    Two Dimensional Elastic Collision

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    If we consider two colliding objects in 2 dimensions, the net or overall velocity of each object should be divided into two perpendicular velocities:
    1. The first one tangent to the common normal surfaces of the colliding objects at the point of colliding bodies having collision in contact.
    2. The second one in alignment of the line of collision.

    Two Dimensional Elastic Collision

    When a force is imparted only in alignment or along the line of collision, the velocities that are tangent to the point of collision are not altered. Also the velocities that are in alignment with the line of collision can be used in the same equations as a one-dimensional collision. There can be final velocities calculation from the newly calculated component velocities. It will depend on the point of collision.

    Coefficient of Restitution

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    When there is a head on Collision between two bodies, the ratio of their relative velocity after collision and their relative velocity before collision is called the Coefficient of restitution.Thus $e$ = $\left|\frac {v_{1} - v_{2}} {u_{1}-u_{2}}\right|$
    For a perfectly elastic collision, the value of e is 1
    If 0 < e < 1, then the collision is inelastic.
    For a perfectly inelastic collision, e = 0.
    If e > 1, then the collision is a super-elastic collision.

    Elastic and Inelastic Collisions

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    The comparison between Elastic and inelastic collision is given below:

    S.No
    ELASTIC COLLISION
    INELASTIC COLLISION
    1 Momentum Conserved Momentum Conserved
    2
    Kinetic energy Conserved
    Kinetic energy not conserved
    3
    Example: Bouncing ball example: Bullet shot in wood
    4 Cannot be Perfect Can be Perfect.

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