Elastic Collision in Two Dimensions - JEE Important Topic

When two objects collide, momentum or kinetic energy is transferred from one to the other. Elastic and inelastic collisions are the two categories into which collisions are divided. It is possible for gas molecules to collide without losing any of their kinetic energy. Additionally, momentum and kinetic energy are still conserved. An elastic collision is the name given to this kind of collision. However, an inelastic collision occurs when kinetic energy is lost or transformed into another form of energy.

Elastic Collision

Ideal gas molecules can collide with one another elastically. A gas is made up of a massive number of perfectly elastic spheres called molecules. These spherical gas molecules travel according to Newton's equations of motion when they are in a condition of Brownian or random motion. Their speeds can range from zero to infinity, and they can move in any direction.

Additionally, each of these molecules occupies a very little volume relative to the volume of the gas because their sizes are smaller than the distance between them. These molecules collide with one another and the container walls in an elastic manner, preserving their momentum and kinetic energy.

Elastic Collision Examples

• The basketball bounces back to your hand when you strike it against the ground. The collision is elastic in this case because the kinetic energy is still conserved.

• Atoms collide in an elastic way, similar to how a pool ball would.

• The meeting of two billiard balls.

• When playing pool or snooker, striking the balls with a stick.

Elastic Collision in Two Dimension

Assuming \[{{F}_{net}}=0\] at the beginning ensures that momentum p is conserved. A collision in which one of the particles is initially at rest is the simplest. A coordinate system with an axis parallel to the particle's speed is the most effective option. Since momentum is conserved, the x- and y-axis components of momentum will likewise be preserved. However, in the chosen coordinate system, ${{p}_{y}}$ is initially equal to zero and ${{p}_{x}}$ equals the momentum of the arriving particle. Both details make the analysis simpler.

A collision in two dimensions when the coordinates are set up so that ${{m}_{2}}$ is initially at rest and ${{v}_{1}}$ is parallel to the x-axis. Because many scattering investigations use a target that is stationary in the lab while particles are scattered from it to identify the constituent particles of the target and how they are bonded together, this coordinate system is occasionally referred to as the laboratory coordinate system. Even though the particles can't always be seen immediately, their starting and terminal velocities can.

Assume that ${{m}_{1}}$ and  ${{m}_{2}}$ are two mass particles in a laboratory frame of reference and that ${{m}_{1}}$ collides with ${{m}_{2}}$ that is initially at rest. Let the velocity of mass ${{m}_{1}}$ before the collision be ${{u}_{1}}$ , and after the collision, it moves with a velocity ${{v}_{1}}$ and is deflected by an angle ${{\theta }_{1}}$  with its incident direction, while ${{m}_{2}}$ moves with a velocity ${{v}_{2}}$ and is deflected by an angle ${{\theta }_{2}}$ with its incident direction.

Elastic Collision in Two Dimensions

For components along the x-axis, the equation of conservation of linear momentum states that

${{m}_{1}}{{u}_{1}}={{m}_{1}}{{v}_{1}}\cos {{\theta }_{1}}+{{m}_{2}}{{v}_{2}}\cos {{\theta }_{2}}$ ---(1)

For components along the y-axis,

$0={{m}_{1}}{{v}_{1}}\sin {{\theta }_{1}}-{{m}_{2}}{{v}_{2}}\sin {{\theta }_{2}}$ ---(2)

X and Y Components in Elastic Collision

According to the principle of kinetic energy conservation,

$\dfrac{1}{2}{{m}_{1}}u_{1}^{2}+\dfrac{1}{2}{{m}_{2}}u_{2}^{2}=\dfrac{1}{2}{{m}_{1}}v_{1}^{2}+\dfrac{1}{2}{{m}_{2}}v_{2}^{2}$  ---(3)

Analysing the above equations reveals that finding values for four unknown quantities ${{v}_{1}},{{v}_{2}},{{\theta }_{1}},{{\theta }_{2}}$ using the above three equations is not possible. Because there are four of them, it is unable to anticipate the variable. The above equation, however, allows us to determine the other variable in a certain way if we measure any one of the variables.

Elastic Collision in Two Dimension Derivation

Elastic collisions, in which all of the kinetic energy between the two bodies is retained, occur when two objects collide. It is a collision between two bodies in which their entire kinetic energy is equal, to put it another way. It could also be two-dimensional or one-dimensional. As a result, fully elastic collisions are unlikely to happen in the real world since some energy exchange, however small, is required. When two bodies collide, their combined kinetic energy remains constant, which is known as an elastic collision. In a perfect, totally elastic collision, there is no net transfer of kinetic energy into other forms, such as heat, noise, or potential energy.

When small objects collide, kinetic energy is converted to potential energy, which is connected to an attractive or repulsive force between the particles (when the particles move against the force, the angle between the force and the velocity is obtuse), and then back to kinetic energy.

The collision formula for completely elastic bodies is applied as (elastic collision in two dimension derivation),

${{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}$

Similar to that, the conservation of total kinetic energy is shown as,

$\dfrac{1}{2}{{m}_{1}}u_{1}^{2}+\dfrac{1}{2}{{m}_{2}}u_{2}^{2}=\dfrac{1}{2}{{m}_{1}}v_{1}^{2}+\dfrac{1}{2}{{m}_{2}}v_{2}^{2}$

These equations can be solved directly to find ${{v}_{1}}$ , ${{v}_{2}}$ when ${{u}_{1}}$, ${{u}_{2}}$ are known.,

${{v}_{1}}=\dfrac{{{m}_{1}}-{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{u}_{1}}+\dfrac{2{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{u}_{2}}$

${{v}_{2}}=\dfrac{2{{m}_{1}}}{{{m}_{1}}-{{m}_{2}}}{{u}_{1}}+\dfrac{{{m}_{2}}-{{m}_{1}}}{{{m}_{1}}+{{m}_{2}}}{{u}_{2}}$

This elastic collision in two dimension formula

If both masses are the same, the answer is simple:

${{v}_{1}}={{u}_{2}}$ and ${{v}_{2}}={{u}_{1}}$

Where, ${{m}_{1}}$ is the mass of 1st body, ${{m}_{2}}$is the mass of 2nd body, ${{u}_{1}}$ is the initial velocity of the 1st body, ${{u}_{2}}$ is the initial velocity of the 2nd body, ${{v}_{1}}$ is the final velocity of the 1st body and ${{v}_{2}}$ is the final velocity of the 2nd body.

Conclusion

Two items collide when they briefly come into contact with one another. To put it another way, a collision is a brief reciprocal interaction between two masses in which the momentum and energy of the masses change. In collisions between two masses, the law of conservation of momentum often holds true, but there may be occasional collisions in which kinetic energy is not conserved. When two bodies meet with one another, it is called an elastic collision because there is no loss of overall kinetic energy. In other terms, it is a collision between two bodies when the sum of their kinetic energies is equal. Although many events or locations allow for elastic collisions, the reality is that inelastic collisions happen most frequently. Download the elastic collision in two dimension PDF for further information.