Head-on Elastic Collision [One Dimensional]

It is that elastic collision in which the colliding bodies move along the same straight line before and after the collision.

Consider two bodies A and B of masses m₁ and m₂ respectively, moving along the same straight line, in the same direction. Let be their respective velocities, such that

The two bodies will collide after some time.

During collision, the bodies will be deformed at the region of contact. So, a part of the kinetic energy will be converted into potential energy. The bodies will regain their original shape due to elasticity. The potential energy will be reconverted into kinetic energy. The bodies will separate and continue to move along the same straight line, in the same direction, but with different velocities.

Applying the law of conservation of momentum,

Total momentum before collision = total momentum after collision

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f (in magnitude)

or m₁ (v_1i - v_1f) = m₂ (v_2f - v_2i) --------------- (i)

Since the collision is elastic, the kinetic energy will be conserved.

Kinetic energy before collision = Kinetic energy after collision

Dividing (ii) by (i), we get

v_1i + v_1f = v_2f + v_2i

or v_1i - v_2i = v_2f - v_1f ------------ (iii)

(v_1i - v_2i) is the magnitude of the relative velocity of A w.r.t B.

(v_2f - v_1f) is the magnitude of the relative velocity of B w.r.t A.

It may be noted that the direction of relative velocity is reversed after the collision.

From equation (iii)

v_2f = v_1i - v_2i + v_1f

From equation (i)

m₁ (v_1i - v_1f) = m₂ (v_1i - v_2i + v_1f - v_2i)

or m₁v_1i - m₁v_1f = m₂v_1i - 2m₂v_2i + m₂v_1f

or -m₁v_1f - m₂v_1f = -m₁v_1i + m₂v_1i - 2m₂v_2i

or (m₁ + m₂) v_1f = (m₁ - m₂) v_1i + 2m₂v_2i

Again, from equation (iii),

v_1f = v_2f - v_1i + v_2i

Substituting this value in equation (i) and simplifying, we get

Special cases

Case (i) when m₁ = m_2, i.e., when both the colliding bodies are of the same mass.

The velocity of body A after collision, is equal to the velocity of body B before the collision.

From equation (v),

The velocity of body B after collision, is equal to the velocity of body A before collision.

To sum up, when two bodies of equal masses suffer one-dimensional elastic collision, they interchange their velocities.

Example, Collision between two billiard balls.

Case (ii) When v_2i = 0 i.e., when the body B is initially at rest.

From equation (iv),

From equation (v),

Now, three special sub-cases arise.

Sub-case (a) When m₂ << m₁ i.e., when the mass of body B is negligible as compared to the mass of body A

In other words, body B is a light body and the body A is a massive body.

From equation (vi),

The velocity of the massive body after collision, is nearly equal to its velocity before collision. In other words, the massive body loses only a very small fraction of its kinetic energy.

From equation (vii),

The velocity of the light body after collision, is nearly double the velocity of the massive body before collision.

To sum up, when a massive body suffers an elastic collision with a stationary light body, there is practically no change in the velocity of the massive body, but the light body acquires a velocity which is nearly double the initial velocity of massive body.

Example: Collision between a fast-moving truck and a stationary drum.

Sub-case (b) When m₁ << m₂ i.e., when the body A is a light body and the body B is a massive body.

From equation (vi),

So, the velocity of the light body after collision, is nearly equal (in magnitude) and opposite to its velocity before collision. In other words, the light body transfers only a small fraction of its kinetic energy to the massive body.

From equation (vii),

So, the velocity of the massive body after collision is very small.

To sum up, when a light body suffers an elastic collision with a stationary massive body, the velocity of the light body is reversed and the massive body remains practically at rest.

Example, A rubber ball thrown against a wall.

Sub-case (c) When m₁ = m_2, i.e., when the colliding bodies are of the same mass.

From equation (vi)

v_1f = 0

In words, the velocity of body A after collision is zero.

From equation (vii),

In words, the velocity of body B after collision, is equal to the velocity of body A before collision.

To sum up, when a body suffers an elastic collision with another body of the same mass at rest, it is stopped dead, while the second body starts moving with the same velocity as that of the first.

In sub-case (c), there is hundred per cent transfer of energy.

Example, Collision between a moving billiard ball and a stationary billiard ball on a billiards table.