Types of Collisions

Collisions are an everyday phenomenon that occurs when two or more objects come into contact with each other, often resulting in a transfer of energy. They are classified into different types based on how energy is conserved or transformed, including elastic, inelastic, and perfectly inelastic collisions, but firstly, we will discuss the concept of collision.

What is Collision?

A collision is an event where a strong force acts for a very short period between two or more bodies.

⇒ A collision is an isolated event; the energy and momentum of the interacting particles change during the collision.

⇒ The collision may occur by actual physical contact of the involved bodies, for instance, the collision between two billiard balls or a ball and a bat. There may be collisions where there is no actual physical contact, such as the collision of alpha particles by a nucleus.

⇒ Any collision is guided by three distinct identifiable stages: Before, during, and after and the particles are independent. Also, after the collision, the force again becomes zero. During the collision, the particles come in contact with each other, therefore, the force of interaction becomes very large.

⇒ The motion of the bodies is guided by the dominating forces. Since, in most of the practical cases, the magnitude of the interacting force is unknown, Newton’s second law of motion can’t be used in such cases. The initial and final velocities can be computed using the law of conservation of momentum.

For instance, consider two bodies with masses m₁ and m₂, moving with velocities u₁ and u₂. They undergo a collision due to the application of an external force 'F_ext' for a small interval of time, and then the final velocities become v₁ and v₂. 

Conservation of Momentum and Energy during a Collision

According to the fundamental laws of physics, certain attributes apply to any type of collision : 

• Momentum conservation: The collision takes place for a very small interval of time, and during this period the average impulsive force causing the collision is significantly larger than the external force acting on the system. Therefore, during a collision, the application of external forces, such as frictional or gravitational forces is not considered into account. This impulsive force is internal, therefore, the total momentum of the system remains constant for all practical purposes. Therefore, it remains conserved throughout the system.

• Energy conservation: According to the law of conservation of energy, energy can neither be destroyed nor created. It can only be transferred from one medium to another. Therefore, the total energy during a collision always remains conserved. Total energy includes all the plausible forms of energy created and destroyed during a collision, such as mechanical energy, internal energy, excitation energy as well as mass-energy.

The various scenarios that can occur include,

• Multiple objects collide in an elastic collision.
• Multiple objects are involved in an inelastic collision.

Types of collision

The main types of collisions are:

1. Elastic collisions: Both momentum and energy are conserved, meaning the objects bounce off each other without losing any energy.
2. Inelastic collisions: Only momentum is conserved, but some of the energy is lost in the form of heat, sound, or other changes. The objects don't necessarily stick together.
3. Perfectly inelastic collisions: In this case, the objects stick together after the collision, and a lot of the energy is lost.
4. Head-on collision and oblique collision.

Elastic Collision in One Dimension

An elastic collision in one dimension happens when two objects bump into each other in a straight line (like cars crashing head-on). After the collision, they bounce off each other without losing any energy.

In simple terms, both objects keep moving after the collision, and none of their energy is lost — it is just transferred between them. Their total energy and total momentum stay the same before and after the collision.

Using,

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

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

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

i.e., 

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

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

Since elastic collisions keep kinetic energy constant, the total kinetic energy before and after the collision remains unchanged.

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

This is the equation that represents the conservation of energy in a two-object elastic collision.

Inelastic Collision in One Dimension

In an inelastic collision, kinetic energy is not conserved. This means some of the energy is lost or changed into other forms of energy, like heat or sound.

Let's take an example where two objects with the same mass and speed collide and stick together after the collision.

In this case, the initial kinetic energy can be calculated using the formula for kinetic energy,

½ mv² + ½ mv² = mv²

Since the objects stick together after the collision, they stop moving and all their energy is used up. In simple terms, after the crash, they do not have any kinetic energy left to move on their own. This type of collision is called a perfectly inelastic collision.

v = (m1v1 + m2v2)/ (m1+m2)

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

Two-Dimensional Collisions

A two-dimensional collision happens when two objects collide and move in two directions, like on a flat surface. Unlike a one-dimensional collision where objects only move back and forth along a straight line, in a two-dimensional collision, objects can move in different directions at the same time—both horizontally and vertically.

For example, imagine two balls hitting each other on a table: one could move to the left and the other might move up after the collision. The collision involves both the x-axis (left-right) and y-axis (up-down), and the direction and speed of the objects change in both directions.

Read More: Collisions in Two Dimensions

Head-on and Oblique collisions

In a head-on collision, two objects collide directly along the same straight line, moving toward each other. Essentially, they hit each other "front to front."

This is the simplest form of collision and usually involves objects moving along the same path but in opposite directions. The change in velocity occurs along the line of impact.

Example: Two cars moving directly toward each other on a road crash into each other.

Oblique Collision:

In an oblique collision, the two objects collide at an angle, not directly head-on. This means they do not hit straight at each other but come together at some angle, leading to a more complex outcome.

The objects can move in different directions after the collision because their velocities change both along the line of impact and perpendicular to it.

Example: Two billiard balls hitting each other at an angle on a pool table.

Conclusion

Collisions help us understand how things move and interact, from tiny particles to big objects. The formulas for different types of collisions explain how momentum is conserved, meaning the total motion in a system stays the same, even if the objects change speed or direction. These concepts are key to understanding how the world works.

Practice Problems on Types of Collisions

Question 1: Let us assume that, s small balls, each of mass m hit a surface elastically each second with a velocity u m/s. Calculate the force witnessed by the surface.

Solution:

Since, there is a rebound of the ball with unmodified velocity, therefore
The change in velocity = 2u 
Mass colliding with the surface =  sm 
Force experienced by the surface is given by,  F=m\frac{dv}{dt}\\ \therefore F=2\ msu

Question 2: Initially, a car of mass 400 kg traveling with a speed of 72 kmph accidentally crashes into a car of mass 4000 kg. Before the collision, the truck was traveling at a speed of 9 kmph, in the same direction as that of the car. As a result of the collision, the car retorts back at a speed of 18 kmph. Calculate the velocity with which the truck is moving after the collision. 

Solution:

By the law of conservation of linear momentum, we have, 
The summation of initial momentum is equivalent to the summation of final momentum. 
m₁u₁ + m₁u₂ = m₁v₁ + m₂v₂
⇒ 400 × 72 + 4000 × 9 = 400 × (-18) + 4000 × v₂
⇒ v₂ = 18 km/h
Therefore, the final velocity of the car is 18 km/h.

Question 3: A crate is kept at the origin of mass equivalent to 5 kg. A force of 10 N is applied to it to simulate its movement at an angle of 60° with the x-axis. As a result of this displacement, the crate moves by 4 m. What is the work done by the application of the force?

Solution:

Work done is given by the dot product of force and displacement, that is, 
F.s = F. s cos θ
=10 × 4 × cos 60° 
= 20 J

Question 4: A crate is kept on a table of mass equivalent to 1 kg. It is dragged along the horizontal table through the length of 1 m by an external force of 8 N. It is then subjected to a vertical height of 2 m. Compute the net work done.

Solution:

Work done to displace the crate horizontally, W_H 
= F × s 
= 8 × 1 = 8 J
Work done to raise the crate vertically, W_v =  F × s 
= m x g x h
 = 1 × 10 × 2 = 20 J
Therefore,
Net work done on the crate is W_H + W_v 
= 8 +20 = 28 J

Question 5: A horizontal force of 5 N is required to maintain a velocity of 2 m/s for a block of 10 kg mass sliding over a rough surface. The work done by this force in one minute is.

Solution:

Mathematically, 
Work done by an object = Force × displacement
⇒ W = F × s
Since, we know, 
s = v x t 
= F × v × t 
= 5 × 2 × 60 
= 600 J

Related Reads,

• Physics
• Collisions in One Dimension
• Difference between Elastic and Inelastic collision