A Brief Note On Types Of Collisions

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To understand the types of collisions, we must first understand what collisions are.

Collisions can be termed as an impact between 2 particles, creating an exchange of forces between those particles in a short time span. In every collision, momentum is always conserved.

The different cases that can occur are as follows:

Multiple objects colliding in an elastic collision 

Multiple objects in an inelastic collision.

Types of Collisions 

The main types of collisions are as follows:

• Elastic collisions: both momentum as well as kinetic energy are conserved
• Inelastic collisions: only momentum is conserved
• Perfectly inelastic collisions: The kinetic energy is lost, resulting in the colliding objects to stick to one another after the collision
• Head-on collision and oblique collision

Elastic Collision in One Dimension

The momentum principle plays a vital role in all cases of two-object collisions. Let us consider a 2-object collision in one dimension. Every elastic collision always conserves kinetic energy, specifically its internal kinetic energy. In other words, the sum of internal kinetic energy is equivalent to the total kinetic energy of the objects in the system.

For an elastic collision to occur in its truest form in subatomic particles, the larger collision may come very close to an elastic collision, but it’s not possible to be exact.

Many factors, such as environmental energies like heat transfer or friction, interfere with larger collisions, resulting in slight alterations.

An example of a nearly perfect elastic collision is a collision that occurs on minimal friction surfaces, like two pucks colliding on an ice surface. In an elastic collision in one dimension, the velocities of both the particles are along the line of collision.

The formulae for problem solutions for one-dimensional elastic collisions are derived from the equation of the conservation of momentum (1) and the conservation of internal kinetic energy (2).

Using (1) 

p1 + p2 = p’1  + p’2    (net external force = 0)

Where p1 and p2 are initial momentum of object 1 and 2 respectively. 

p’1 and p’2 are the final momentum of object 1 and 2 respectively.

i.e., 

m1v1 + m2v2 = m1v’1 + m2v’2     (net force = 0)

Here, p denotes the momentum of the particles, m the mass, and v the velocities of the particles.

Since all elastic collisions conserve kinetic energy, the sum of kinetic energy before and after the collision will be the same. Therefore, 

½( m1v1²  + m2v2²)    =  ½( m1v’1²  + m2v’2²) 

This is the equation for conservation of energy for a 2-object elastic collision.

Inelastic Collision in One Dimension

In an inelastic collision, the internal kinetic energy is not constant, i.e., it is not conserved.

This means that the collision leads to some loss of kinetic energy or converts kinetic energy to another form of energy.

Let’s take an example where two objects having the same mass and speed collide with each other but stick to one another after a collision.

Then, their initial kinetic energy is:

½ mv² + ½ mv² = mv²

Since they stick together after the collision, the net internal kinetic energy becomes 0 (this is also called a perfectly inelastic collision)

The formula used for inelastic collisions is:

v = (m1v1 + m2v2)/ (m1+m2)

Here, m stands for mass and v for velocities of the respective particles.

Two-Dimensional Collisions

In this article, we have covered all aspects of collisions, such as momentum and how it is conserved and kinetic energy and how it transforms in elastic and inelastic collisions, but these are mainly on account of one-dimensional collisions. What are two-dimensional collisions?

If 2 balls were to collide, there are two ways the balls can bounce back. 

1. They bounce back in the same direction they collided, i.e., in a single dimension
2. The more likely outcome is that they collide in two different directions, becoming a 2-dimensional figure

Just as in one dimension, the momentum is conserved even in two dimensions. 

Also, it can be expressed in x and y directions. 

X direction 

Summation of momentum before collision = x

Summation of momentum after collision = x’

X = X’

Y direction 

Similarly, summation of momentum before collision = y

Summation of momentum after collision = y’

Y = y’

Here, the collisions are also factoring in momentum along with kinetic energy and momentum.

Even in two-dimensional collisions, the kinetic energy is conserved in elastic collisions.

The summation of kinetic energy before collision (½ mv²) is equal to the summation of kinetic energy after collision (½ mv’²).

½ mv² = ½mv’²

There are two primary cases in which we can use two-dimensional collisions: stationary and moving.

In this case, taking into account all vector information and momentum,

x1 is the angle of object one, and x2 is the angle of object 2.

tan x1 =( m2 tanx) / ( m1 + m2 cosx)

x2 = – x1/ 2 

Therefore, velocity (in terms of magnitude is) 

v1’ = v1((m1 + m2 + 2m1m2cosx)½ / (m1 + m2)

v2’ = v1((2m1/(m1+m2))sin(x/2)

Head-on and Oblique collisions

Head-on Collisions 

When collisions of objects occur in a straight line or when the centres of gravity of each of the objects are connected, it is a head-on collision. The angle of impact is always 90°.

Head-on collisions may occur in various ways. 

Case1 

m1 = m2 and object 2 is stationary, then 

v’1 (final velocity of object 1) = 0, and final velocity of object 2 v’2 = v1 (initial velocity of object 1)

Case 2:

m1 >> m2 

The velocity of a stationary particle will be 2 times the initial velocity of an object in motion 

i.e., v’1 = v1 and v’2 = 2(v1)

Case 3:

 m1 << m2 

If the mass of the object in motion is much smaller than the stationary object, then the final velocity of both the objects will become zero 

i.e.,  v’1 = 0 and v’2 = 0

Conclusion

These cover all the types of collisions and their formulae. It is important to understand how the world around us works, and collisions play an essential role in doing so from the microscopic and macroscopic standpoints. These formulae help in an better understanding of conservation of momentum also.