Elastic and Inelastic Collisions in one and two Dimensions

In physics, collisions are a type of momentum and we must note that in all types of collisions, momentum is always conserved.

Types of collision

The three main types of collisions are:

• ELASTIC COLLISIONS: momentum as well as kinetic energy is conserved 

• INELASTIC COLLISIONS: momentum is conserved.

• PERFECTLY INELASTIC COLLISIONS: the kinetic energy is lost, resulting in the colliding objects sticking to one another after the collision.

Elastic collision in one dimension

In all 2 object collisions, the momentum principle plays a vital role. Let us consider a 2 object collision in one dimension, every elastic collision always conserves its kinetic energy in specific its internal kinetic energy, or the sum of internal kinetic energy is equivalent to the total kinetic energy in the objects in the system.

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

In larger collison many factors such as environmental energies like heat transfer or friction interfere with the collision resulting in slight alterations.

Example of nearly perfect elastic collisions are collisions that occur on minimal friction surfaces like 2 plucks colliding on an ice surface.

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

Using (1) 

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

i.e, 

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

Since all elastic collisions conserve kinetic energy the sum of kinetic energy is equal before and after collision hence 

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

This equation is 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 in kinetic energy or converts kinetic energy to another form of energy.

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

Then their initial kinetic energy is 

½ mv² + ½ mv² = mv²

Since they stick together after collision the net internal kinetic energy becomes 0. (also called perfectly inelastic collision)  

The formula used for inelastic collisions 

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

Two dimensional collisions

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

What are two-dimensional collisions?

If 2 balls were to collide there are 2 ways the balls bounce back 

1- they bounce back in the same direction they collided in i.e in a single dimension

2- the more likely outcome is that they collide in 2 different directions hence becoming a 2 dimensional figure.

Just as in one dimension the momentum is conserved even in the second dimension.

Also 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 angular momentum along with kinetic energy and momentum.

Even in 2-dimensional collisions the kinetic energy is conserved in elastic collisions.

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 2 primary cases we can use during 2-d collisions 

One stationary and one moving.

In this case, taking in account all vector information and angular momentum

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

tan x1 =( m2 tanx1) / ( m1 + m2 cosx1)

x2 = 𝜋 – x1/ 2 

Hence velocity (in terms of magnitude is ) 

v2 = v1 (m1 + m2 + 2m1m2 cos x1)½ / (m1 + m2)

v2 = v1((2m1/(m1+m2))sin(x1/2)

Elastic collision in 2 dimension 

Assuming 2 objects a1 and a2 are moving towards each other with velocity v1 and v2.

If the resulting collision is at an angle say x

To solve this we will use momentum conservation and kinetic energy laws for the formula 

Kinetic energy :

Here since energy is one dimensional the total is a scalar quantity 

½ mv1² + ½ mv2² = ½ mv1’² + ½ mv2’²

In the momentum aspect of solving we have to consider angle or direction of the force

We can do so by splitting the components into x and y 

In x component apply conservation of linear momentum

pox = pfx

m1v1-m2v2 = m1’v1’fcosθ1’ + m2’v2’fcosθ2’ 

Similarly in terms of y, initially due to both only moving in the x component there is no initial momentum in y, the final momentum is found after the collision but using trigonometry we get 

py = pfy

0 =  m1v1’fsinθ1’ + m2’v2’fsinθ2’

Inelastic collision in 2D

Let us take an example of inelastic collision and its tragectory 

Here similar to elastic collision the vectors and variables must be accounted for.

Generally in inelastic collisions, m1 and m2 are objects moving towards one another at an angle of x degrees moving with v1 and v2 speed.

If they undergo inelastic collision there are x1 and x2 angles and form a single mass velocity M with velocity vf 

Since inelastic collision energy cannot be conserved we have to use conservation of momentum to solve any questions.

Splitting the components into 2 , we get the x and y component 

x component:m1v1 + m2v2cosθ1=Mvfcosθ2 

y component:m2v2sinθ1=Mvfsinθ2

Since only 2 unknown values are present , the formula is universal to all equations of inelastic collision in 2 dimensions.

Conclusion

In physics, collisions are a type of momentum and we must note that in all types of collisions, momentum is always conserved. Let us consider a 2 object collision in one dimension, every elastic collision always conserves its kinetic energy in specific its internal kinetic energy, or the sum of internal kinetic energy is equivalent to the total kinetic energy in the objects in the system.