All About Elastic Collision Examples

Elastic collisions are when 2 objects collide or impact, creating a mutual force exertion between the 2 entities.

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

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, specifically its internal kinetic energy.

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

Examples of Elastic Collision

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

Examples of nearly perfect elastic collisions are collisions that occur on minimal friction surfaces, like 2 pucks colliding on an ice surface.

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

The formula used to calculate elastic collisions, when momentum is conserved, is as follows:

Using (1) 

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

i.e., 

M₁v₁ + m₂v₂ = m₁v’₁ + m₂v’₂     (net force = 0)

 Formula when kinetic energy is conserved 

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

Perfect Elastic Collisions 

When the collision results in no loss of kinetic energy, it is known as a perfectly elastic collision.

It is nearly impossible for us to achieve this in our day-to-day life as it requires there to be no external forces like friction, etc.; therefore, perfectly elastic collisions always take place in ideal conditions.

Approximating Perfectly Elastic Collisions

Even though there is no perfect elastic collision in real life, we think there is no point in learning the concept; however, it is very useful in calculations. This is due to the fact that kinetic energy is always conserved, which acts as a constraint in calculations. If we were to not assume they are nearly perfect, we will have to address many unresolved questions. We can consider then that the actual value is not varying on a large scale; in fact, it is quite small.

Let us consider a case of 2 balls colliding head on; the 2 balls a and b have only their initial velocity values va and vb, and we want to find their final velocity.

We apply the law of conservation of momentum, and we can get this equation:

M_a v_a + m_b v_b = m_a  v’_a + m_b v’_b 

Since kinetic energy is also conserved, we get 

½  M_a (v_a )² +½  M_b (v_b )² = ½  M_a (v’_a )² +½  M_b (v’_b )²

Solving these 2 constraints using simultaneous equations to get the final velocities of both objects, we get     

V’_a = ((m_a – m_b)/ m_a+ m_b)) v_a + (2m_b/(m_a + m_b)) v_b 

V’_b =  (2m_a/(m_a + m_b)) v_a + ((m_b – m_a)/ m_a+ m_b)) v_b

Note: These equations are always limited to certain cases of head-on collisions. They help us understand the physics behind the collisions.

Case 2: If one object is at rest while the other is in motion collide.

Here, say ball y collides with ball x (at rest) 

Final velocity of y becomes 0, and x gains the speed of y

i.e.,

V’_y = 0 and v’_x = v_y (initial speed of y )

The 2 objects bounce off each other, exchanging velocity. This result is unique, as it is only viable for objects colliding with equal momentum but in opposite directions.

Classification of Elastic Collision

One-Dimensional Elastic Collision

Collisions can also be classified based on the number of dimensions involved in measuring the collision. When a collision occurs such that all the variables that are involved in the collision can be measured in a single dimension, it is called a collision in one dimension. In simple words, when the initial and final velocities of the masses that are colliding both lie on one, it is called a collision in one dimension.

During an elastic collision, both the momentum of the bodies and their kinetic energies are conserved. When you consider bodies 1 and 2  with masses m₁ and m₂, respectively, and velocities calculated before the collision and v₁ and v₂ calculated after the collision. Therefore, the conservation of momentum that occurs pre and post the collision is expressed through:

m₁u₁+m₂u₂ = m₁v₁+m₂v₂