# Collisions

COLLISION

A collision occurs when two objects come in direct contact. It is the event in which two or more bodies exert forces on each other in about a relatively short time.

There are two types of collisions:

1. Inelastic collisions: momentum is conserved,
2. Elastic collisions: momentum is conserved and kinetic energy is conserved.

Elastic Collision

An elastic collision is one where there is no net loss in kinetic energy in the system due to the collision.

Inelastic Collision

An inelastic collision is a type of collision where this is a loss of kinetic energy. The lost kinetic energy is transformed into thermal energy, sound energy, and material deformation.

## Elastic Collision

When two bodies collide but there is no loss in the overall kinetic energy, it is called a perfectly elastic collision

### Elastic Collision Definition:

An elastic collision is a collision in which there is no net loss in kinetic energy in the system due to the collision. Both momentum and kinetic energy are conserved in an elastic collision.

Basically in the case of elastic collision, the kinetic energy before and after the collision remains the same and is not converted to any other form of energy.

It can be either one-dimensional or two-dimensional. In the real world, perfectly elastic collision is impossible because there is bound to be some energy conversion, however small.

However, though there is no change in the linear momentum of the whole system, there is a change in the individual momenta of the involved components, which are equal and opposite in magnitude and cancel each other out and the initial energy is conserved.

The collision of billiard balls is nearly elastic because the kinetic energy is conserved before and after the collision

### Elastic Collision Examples

• When a ball at a billiard table hits another ball, it is an example of elastic collision.
• When you throw a ball on the ground and it bounces back to your hand, there is no net change in the kinetic energy, and hence, it is an elastic collision.

## Elastic Collision Formula

The Elastic Collision formula of momentum is given by:

m1u1 + m2u2 = m1v1 + m2v2

Where,

• m1 = Mass of 1st body
• m2 = Mass of 2nd body
• u1 =Initial velocity of 1st body
• u2 = Initial velocity of the second body
• v1 = Final velocity of the first body
• v2 = Final velocity of the second body

The Elastic Collision formula of kinetic energy is given by:

(1/2) m1u12 + (1/2) m2u22 = (1/2) m1v12 + (1/2) m2v22

# Inelastic Collision

In physics, an inelastic collision occurs when some amount of kinetic energy of a colliding object/system is lost. The colliding particles stick together, and the maximum amount of kinetic energy is lost in a perfectly inelastic collision. In such cases, kinetic energy lost is used in bonding the two bodies together. Problems involving collisions are usually solved using the conservation of momentum and energy.

## Inelastic Collision Definition

An inelastic collision is such a type of collision that takes place between two objects in which some energy is lost. In the case of inelastic collision, momentum is conserved but the kinetic energy is not conserved. Most of the collisions in daily life are inelastic in nature.

The above schematic diagram illustrates a perfectly inelastic collision.

What is a collision? A collision is an event in which two or more objects exert forces on each other for a short interval of time. It is categorised into two types:

• Inelastic collision
• Elastic collision

### Perfectly Inelastic Collision

The special case of inelastic collision is known as a perfectly inelastic collision. Here, two objects stick together after collision and move as a single object. Refer to the figure above. For example, when a wet mudball is thrown against a wall, the mudball sticks to the wall.

## Inelastic Collision Formula

When two objects collide under inelastic conditions, the final velocity with which the object moves is given by-

Where,

• V= Final velocity
• M1= mass of the first object in kgs
• M2= mas of the second object in kgs
• V1= initial velocity of the first object in m/s
• V2= initial velocity of the second object in m/s

### Inelastic Collision in Two Dimension

For inelastic collision in two dimensions, conservation of momentum is applied separately along each axis. Since Momentum is a vector equation, there is one conservation of momentum equation per dimension. There is only one conservation of energy equation.

## Inelastic Collision Examples

Most of the collision we see in our day-to-day life falls under inelastic collision. Some of them are listed below.

Real-World Examples of Inelastic Collision

• The ball is dropped from a certain height and it is unable to rise to its original height.
• When a soft mudball is thrown against the wall, it will stick to the wall.
• The accident of two vehicles
• A car hitting a tree

## Inelastic Collision Kinetic Energy

In the case of inelastic collision, the kinetic energy is not conserved. The loss of kinetic energy is due to internal friction. It may turn into vibrational energy of the atoms, causing a heating effect and the bodies are deformed.

An animation of an elastic collision between balls can be seen by watching this video. It replicates the elastic collisions between balls of varying masses.

Perfectly elastic collisions can happen only with subatomic particles. Everyday observable examples of perfectly elastic collisions don’t exist—some kinetic energy is always lost, as it is converted into heat transfer due to friction. However, collisions between everyday objects are almost perfectly elastic when they occur with objects and surfaces that are nearly frictionless, such as with two steel blocks on ice.

Now, to solve problems involving one-dimensional elastic collisions between two objects, we can use the equation for the conservation of momentum. First, the equation for the conservation of momentum for two objects in a one-dimensional collision is

Substituting the definition of momentum p = mv for each initial and final momentum, we get

where the primes (') indicate values after the collision; In some texts, you may see i for initial (before collision) and f for final (after collision). The equation assumes that the mass of each object does not change during the collision.

### Elastic Collision Example Problem

Two billiard balls collide. Ball 1 moves with a velocity of 6 m/s, and ball 2 is at rest. After the collision, ball 1 comes to a complete stop. What is the velocity of ball 2 after the collision? Is this collision elastic or inelastic? The mass of each ball is 0.20 kg.

Solution:
To find the velocity of ball 2, use a momentum table.

 Objects Momentum Before Momentum After Ball 1 0.20 kg × 6 m/s = 1.2 0 Ball 2 0 0.20 kg × v2 Total 1.2 kg × m/s 0.20 kg × v2

1.2 kg × m/s = 0.20 kg × v2

v2 =1.2 / 0.20 = 6 m/s

To determine whether the collision is elastic or inelastic, calculate the total kinetic energy of the system both before and after the collision.

 Objects KE Before (J) KE After (J) Ball 1 0.50 × 0.20 × 62 = 3.6 0 Ball 2 0 0.50 × 0.20 × 62 = 3.6 Total 3.6 3.6

Since the kinetic energy before the collision equals the kinetic energy after the collision (kinetic energy is conserved), this is an elastic collision.

### Inelastic Collision

The total kinetic energy is conserved.

The total kinetic energy of the bodies at the beginning and the end of the collision is different.

Momentum is conserved.

Momentum is conserved.

No conversion of energy takes place.

Kinetic energy is changed into other energy such as sound or heat energy.

Highly unlikely in the real world as there is almost always a change in energy.

This is the normal form of collision in the real world.

An example of this can be swinging balls or a spacecraft flying near a planet but not getting affected by its gravity in the end.

An example of an inelastic collision can be the collision of two cars.

## Applications of Elastic Collision

• The collision time affects the amount of force an object experiences during a collision. The greater the collision time, the smaller the force acting upon the object. Thus, to maximize the force experienced by an object during a collision, the collision time must be decreased.
• Likewise, the collision time must be increased to minimise the force. There are several real-world applications of these phenomena. The airbags in automobiles increase the collapse time and minimize the effect of force on objects during a collision. Airbag accomplishes this by extending the time required to stop the momentum of the passenger and the driver.

# Law of Conservation of Linear Momentum

The linear momentum of a particle is defined as the product of the mass of the particle times the velocity of that particle. Conservation of momentum of a particle is a property exhibited by any particle where the total amount of momentum never changes. Linear momentum of a particle is a vector quantity and is denoted by

## Conservation of Linear Momentum

According to the conservation of linear momentum,

If the net external force acting on a system of bodies is zero, then the momentum of the system remains constant.

We have to remember that the momentum of the system is conserved and not that of the individual particles. The momentum of the individual bodies in the system might increase or decrease according to the situation, but the momentum of the system will always be conserved, as long as there is no external net force acting on it.

## Conservation of Linear Momentum Formula

The principle of conservation of momentum states that if two objects collide, then the total momentum before and after the collision will be the same if there is no external force acting on the colliding objects.

The conservation of linear momentum formula mathematically expresses that the momentum of the system remains constant when the net external force is zero.
Initial momentum = Final momentum

 Pi = Pf

## Conservation of Linear Momentum Equation

The law of conservation of momentum can be explained from the second law of motion. Newton’s second law of motion says that the rate of change of linear momentum of a body is equal to the net external force applied to it.

Mathematically it is expressed as:

If the net external force acting on a body is zero, then the rate of change of momentum is also zero, which means that there is no change in momentum.

## Conservation of Linear Momentum Example

Two bodies of mass M and m are moving in opposite directions with the velocities v. If they collide and move together after the collision, we have to find the velocity of the system.

Since there is no external force acting on the system of two bodies, momentum will be conserved.

Initial momentum = Final momentum

(Mv – mv) = (M+m)VFinal

From this equation, we can easily find the final velocity of the system.

## Conservation of Linear Momentum Applications

One of the applications of conservation of momentum is the launching of rockets. The rocket fuel burns are pushes the exhaust gases downwards, and due to this, the rocket gets pushed upwards. Motorboats also work on the same principle, it pushes the water backward and gets pushed forward in reaction to conserve momentum.

# Law of Conservation of Momentum Derivation

The law of conservation of momentum is one of the most prominent laws in physics. The conservation of momentum law principle tells us that the total momentum of a system is always conserved for an isolated system. Let us learn more about the conservation of momentum along with derivation.

## Momentum Conservation Principle

Law of conservation of momentum states that

For two or more bodies in an isolated system acting upon each other, their total momentum remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed.

The principle of conservation of momentum is a direct consequence of Newton’s third law of motion.

## Examples of Law of Conservation of Momentum

Following are the examples of law of conservation of momentum:

• Air-filled balloons
• System of gun and bullet
• Motion of rockets