Collisions

The collision is the mutual interaction between two particles for a short interval of time so that their momentum and kinetic energy may change. A collision is an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time. Actual physical contact is not necessary for a collision.

There are two types of collisions: (i) Elastic collision and (ii) Inelastic collision

Elastic Collision

If the kinetic energy and momentum are conserved in the collision, it is said to be the elastic collision. In the elastic collision, the nature of the force is conservative. Collisions between atomic or sub-atomic particles, between gas molecules etc. are the perfectly elastic collision.

Characteristics of Elastic Collision

1. The momentum is conserved.
2. Kinetic energy is conserved.
3. Total energy is conserved.
4. Forces involved in the interaction are conservative in nature.
5. Mechanical energy is not converted into other forms of energy.

Elastic Collision in One Dimension

If the colliding bodies move along the same straight path before and after the collision, it is said to be one-dimensional collision

Consider two objects of masses m₁ and m₂ moving with velocities u₁ and u₂ such that u₁>u₂ in the same path, collide and let after collision their velocities be v₁ and v₂ on the same line as shown in the figure.

From the principle of conservation of momentum

m₁u₁+m₂u₂ =m₁v₁+mv₂ ............. (i)

m₁(u₁-v₁) =m₂ (v₂- u₂) ..................... (ii)

since the collision is elastic

K.E. before collision = K.E after collision

or \(\frac{1}{2}\)m₁u₁² + \(\frac{1}{2}\) m₂u₂² = \(\frac{1}{2}\)m₁v₁²+ \(\frac{1}{2}\)m₂v₂²

m₁(u₁²-v₁²) =m₂ (v₂²- u²₂)

m₁(u₁-v₁) (u₁+v₁) =m₂ (v₂- u₂ )m₂ (v₂+u₂) .............................(iii)

From equation (iv), it follows that relative velocity of approach (u_1­-u₂) before collision is equal to the relative velocity of separation (v₂-v₁) after collision. From equation (iv)

\begin{align*} v_2 &= u_1 –u_2 + v_1 \\ \text {Substituting the value of} v_1 in \text {equation} (i) \text {we get} \\ v_1 = v_2 –u_1 + u_2 \\ \text {Substituting for value of } v_1 in \text {equation}, (i), \text {we get} \\ m_1u_1 + m_2u_2 &= m_1(v_2 –u_1+u_2) + m_2v_2 \\ (m_2-m_1)u_2 + 2m_1u_1 &= (m_1 + m_2) v_2 \\ \text {or,} v_2 &= \frac {(m_2 –m_1) u_2 + 2m_1u_1 }{m_1 + m_2} \dots (iv)\\ \end{align*}

Special Cases

1. When m₁ = m₂, then from the equation (v) and (vi), we have
   $$v_1 = u_2 \text {and} v_2 = u_1 $$
   e. if two bodies of equal masses suffer elastic collision, then after the collision they will interchange their velocities.
2. When u₂=0 i.e. when 2^nd body is at rest then from equation (v), we have
   $$v_1 =\frac {m_1 –m_2}{m_1+ m_2} u_1 \dots (vii) $$
   and from equation (vi), we have
   $$v_2 =\frac {2m_1}{m_1+m_2} u_1 \dots (viii) $$

• When m₁>>m₂ and u₂ = 0, then equation (vii) and (viii) give
  $$v_1 = u_1 \text {and} v_2 = 2u_1 $$
  e. velocity of massive body is same but the lighter body acquires a velocity which is double the initial velocity of massive body.
• When m₂>>m₁ and u₂ = 0, then equation (vii) and (viii) give
  $$v_1 \approx –u_1 \text {and} v_2 \approx 0$$
  e. the velocity of the lighter body is reversed and massive remains at rest.

Inelastic Collision

The collision in which momentum is conserved but K.E is not conserved called inelastic collision. In this collision, the nature of force is nonconservative. In the inelastic collision, kinetic energy is lost in the form of heat energy, sound energy, light energy etc.

In the case of inelastic collision between large particles, the loss of kinetic energy occurs mostly in the form of heat energy due to increased vibrations of the constituent atoms of the particles. So in the case of elastic atomic collisions if they are in the normal state before the collision and they become excite after collision i.e. some kinetic energy at the collision is converted into excited energy called excitation energy. Hence K.E. before collision is always greater than after collision i.e.

$$\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2 +\frac{1}{2}m_2v_2^2+\xi$$

If the particles are in excited state before collision and carries into normal state after collision, then final K.E. will be greater than the initial K.E. i.e.

$$\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2 +\frac{1}{2}m_2v_2^2+\xi$$

Characteristics of Inelastic Collision

1. The momentum is conserved.
2. Total energy is conserved.
3. Kinetic energy is not conserved.
4. Mechanical energy is converted into other forms of energy.
5. Forces involved during interaction are non-conservative in nature.

Inelastic Collision in One Dimension

Let us consider two perfectly inelastic bodies A and B of mass m₁ and m₂. Body A is moving with velocity u₁ and B is at rest. After some time they collide and move together with common velocity v.

\begin{align*} \text {So initial momentum before collision} &= m_1u_1 \\ \text {Final momentum before collision} &= m_1u_1 \\ \text {Final momentum after collision} &= (m_1 + m_2) v \\ \therefore v &= \frac {m_1u_1}{m_1 + m_2} \dots (i) \\ \frac {\text {K.E. before collision}}{\text {K.E. after collision}} &= \frac {\frac 12 m_1u_1^2}{\frac 12 (m_1 + m_2)v^2 } &= \frac {m_1u_1^2}{(m_1+m_2) \left (\frac {m_1v_1}{m_1+m_2}\right )^2 \\ &= \frac {m_1+m_2}{m_1} >1 \\ \therefore \text {K.E. before collision}=\text {K.E. after collision} \end{align*}

Coefficient of Restitution

The ratio of relative velocity of separation and relative velocity of approach is a constant and this constant is called coefficient of restitution. It is denoted by e.

$$ e = \frac {v_2 –v_1}{u_1-u_2} $$

1. For perfectly elastic collision, e=1
2. For perfectly inelastic collision, e=o
3. For super elastic collision e>1
   but in general, we have 0<e<1.