## What is Collision?

Collision is a very common phenomenon, which we encounter in everyday life. We all might have played games like cricket, in which the batsman hits a ball which is thrown towards him. This is an example of collision, the collision takes place between the bat and the ball as the two bodies hit each other. In physics, collision is when two different bodies exert forces on each other in a small duration of time. Collision is an important topic in physics.

## The physics of collisions

In cricket, when the fast-moving ball is made to hit with a bat, the ball undergoes collision. After the collision, not only the direction is changed but also the speed of the colliding object is changed. If two bodies moving with certain velocities hit each other, both the speed of the colliding objects and direction change.

We know that any moving object has some momentum and kinetic energy. Momentum is the product of mass and speed. Any moving body possesses momentum. In an example of two bodies that are about to undergo collision, if the two masses travel with a certain speed towards each other, with a certain speed. After the bodies undergo collision, the momentum of that system remains the same. The momentum remains the same throughout. The moving object not only possesses momentum but also kinetic energy. The kinetic energy of the collision does not remain the same. Sometimes, it is conserved and for some cases, it is not conserved. Based on this concept, the collision is classified into different types.

## Types of collisions

Collisions are mainly classified based on the conservation of kinetic energy as,

- Elastic collision.
- Inelastic collision.

Consider the example of the collision of metal balls. As two metal balls undergo collision, both the direction, as well as the speed of the balls change. But the net kinetic energy of the metal balls remain the same. Hence, the kinetic energy remains the same.

Now, consider an example in which a clay ball is hit on a wall, which is also regarded as a collision. After the collision, we can observe that the ball completely loses its energy, and the speed of the ball is reduced drastically.

**Note 1:**kinetic energy is lost in an inelastic collision but the net energy of the colliding system remains the same. The kinetic energy is converted to other forms of energy; the energy is used to strain the colliding object.**Note 2:**In reality, the collisions are neither perfectly elastic nor perfectly inelastic. Even in the elastic collision, some fraction of kinetic energy is lost as light and sound. Also, energy is lost due to friction.

### Elastic collision

Let us discuss the collision of a very simple type since the two-body collision happens along a straight line. Consider the example of the collision of the metal balls, assume that the two bodies travel towards each other at a certain velocity. The collision along a line is called a one-dimensional collision. Let *m *and *m’ *be the masses of the two bodies, *u *and *u’* be the initial velocities.

The initial momentum is,

${p}_{i}=mu+m\text{'}u\text{'}$

The initial kinetic energy is,

${K}_{i}=\frac{1}{2}m{u}^{2}+\frac{1}{2}m\text{'}u{\text{'}}^{2}$

Let v and v' be the final velocities after the collision.

Hence, the final momentum is,

${p}_{f}=mv+m\text{'}v\text{'}$

The final kinetic energy is,

${K}_{f}=\frac{1}{2}m{v}^{2}+\frac{1}{2}m\text{'}v{\text{'}}^{2}$

From laws of conservation, the momentum of any collision remains conserved.

So,

${p}_{i}={p}_{f}$

$mu+m\text{'}u\text{'}=mv+m\text{'}v\text{'}$ (i)

In our example, the collision is perfectly elastic. So, the kinetic energy is not lost. It is said that the total kinetic energy of the system is conserved.

Hence, we can arrive,

${K}_{i}={K}_{f}$

$\frac{1}{2}m{u}^{2}+\frac{1}{2}m\text{'}u{\text{'}}^{2}=\frac{1}{2}m{v}^{2}+\frac{1}{2}m\text{'}v{\text{'}}^{2}$

$m{u}^{2}+m\text{'}u{\text{'}}^{2}=m{v}^{2}+m\text{'}v{\text{'}}^{2}$ (ii)

The problem is much simpler if the masses of two objects are the same.

i.e. $m=m\text{'}$

Equation (i) becomes,

$mu+mu\text{'}=mv+mv\text{'}$

$m(u+u\text{'})=m(v+v\text{'})$

$u+u\text{'}=v+v\text{'}$ (iii)

Equation (ii) becomes,

$m{u}^{2}+mu{\text{'}}^{2}=m{v}^{2}+mv{\text{'}}^{2}$

$m({u}^{2}+u{\text{'}}^{2})=m({v}^{2}+v{\text{'}}^{2})$

${v}^{2}={u}^{2}+u{\text{'}}^{2}-v{\text{'}}^{2}$ (iv)

As the first object was at initially rest, the equation (iii) and (iv) becomes,

$v\text{'}=u\text{'}+v$ (v)

${v}^{2}=u{\text{'}}^{2}-v{\text{'}}^{2}$

Substituting equation (v) in (iv),

${v}^{2}={u}^{2}-v{\text{'}}^{2}$

$v{\text{'}}^{2}={u}^{2}-{(u\text{'}+v)}^{2}$

$v{\text{'}}^{2}={u}^{2}-u{\text{'}}^{2}-v{\text{'}}^{2}-2vu\text{'}$

$2{v}^{2}=-2vu\text{'}$

${v}^{2}+vu\text{'}=0$

By solving the above equation we get,

$v=0$

and v'=u'

This means the kinetic energy of the colliding object is completely transferred to the second object.

The final velocity for the collision of objects of different masses is,

$v\text{'}=\frac{m-m\text{'}}{m+m\text{'}}u\frac{2m}{m+m\text{'}}u\text{'}$

### Inelastic collisions

In the above example, the kinetic energy of the metal ball system is constant. In simple words, the total kinetic energy remains constant. Consider an example in which clay is hit on a rigid wall. If the clay is hit at a certain speed, the clay gets attached to the surface. After the collision, the object remains at rest. Hence, the kinetic energy of the collision is completely lost. The colliding bodies, after collision move with common speed. These collisions are termed inelastic collisions.

Consider the collision of masses *m* and *m’*. The collision is said to be inelastic. The initial velocities before the collision are *u* and *u’*.

Hence the initial net momentum is,

${p}_{i}=mu+mu\text{'}$

If the collision is perfectly inelastic then colliding bodies get attached and travel with a common velocity v.

Hence the final momentum is,

${p}_{f}=(m+m\text{'})v$

The following equation is valid as the net momentum of this system is conserved.

$mu+mu\text{'}=(m+m\text{'})v$

The final velocity is,

$v=\frac{mu+m\text{'}u\text{'}}{m+m\text{'}}$

## Formulas

For elastic collision the final speed is,

$v\text{'}=\frac{m-m\text{'}}{m+m\text{'}}u+\frac{2m\text{'}}{m+m\text{'}}u\text{'}$ (in m/s)

The final velocity of inelastic collision is,

$v=\frac{mu+m\text{'}u\text{'}}{m+m\text{'}}$ (m/s)

Where,

u is the initial velocity.

v is the final velocity.

## Context and Applications

This topic is significant in physics for both undergraduate and graduate courses, especially for bachelors and masters in science (physics).

## Practice Problems

**Question 1: **An object of mass 5 kg hits a block with a speed of 100 m/s. The mass of the block was estimated to be about 10 kg. The block was initially at rest. The surface used was frictionless. What is the speed after collision?

(a) 55.8 m/s

(b) 33.33 m/s

(c) 66.79 m/s

(d) 98.96 m/s

**Answer: **The correct option is (b).

**Given data:**

$m=10\mathrm{kg}$

$m\text{'}=5kg$

$u=100m/s$

$u\text{'}=0m/s$

**Explanation:**

The final speed is,

$v\text{'}=\frac{m-m\text{'}}{m+m\text{'}}u+\frac{2m\text{'}}{m+m\text{'}}u\text{'}$

$v\text{'}=\frac{10-5}{10+5}\times 100m/s+\frac{2\times 5}{10+5}\times 0m/s$

$v\text{'}=33.33m/s$

The final speed is 33.33 m/s.

**Question 2: **Which of the following is conserved for inelastic collision?

(a) Potential energy

(b) Kinetic energy

(c) Momentum

(d) Acceleration

**Answer**: The correct option is (c).

**Explanation: **For any collision, the momentum is conserved. For inelastic collision, the kinetic energy is not conserved. Hence, the correct answer is momentum.

**Question 3: **The collision of two metal balls is an example of ____.

(a) Inelastic collision

(b) Elastic collision

(c) Two-dimensional collision

(d) None of the above

**Answer: **The correct option is (b).

**Explanation: **During the collision of two metal balls both kinetic energy and momentum are conserved. Hence, the collision is elastic.

**Question 4: **Which of the following quantity is completely lost in a perfectly inelastic collision?

(a) Potential energy

(b) Kinetic energy

(c) Momentum

(d) Total energy

**Answer:** The correct option is (b).

**Explanation: **If the collision is perfectly inelastic then the final velocity of the colliding bodies is zero as the kinetic energy is lost completely.

**Question 5: **The physics of collision is based on the following concepts ____.

(a) Conservation of kinetic energy

(b) Conservation of momentum

(c) Conservation of matter

(d) Both (a) and (b)

**Answer:** The correct option is (d).

**Explanation: **The collision is explained using the conservation of kinetic energy and momentum.

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