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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

m₁v₁_i + m₂v₂_i = m₁v₁_f + m₂v₂_f (in magnitude)

or m₁ (v₁_i - v₁_f) = m₂ (v₂_f - v₂_i) --------------- (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

v₁_i + v₁_f = v₂_f + v₂_i

or v₁_i - v₂_i = v₂_f - v₁_f ------------ (iii)

(v₁_i - v₂_i) is the magnitude of the relative velocity of A w.r.t B.

(v₂_f - v₁_f) 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)

v₂_f = v₁_i - v₂_i + v₁_f

From equation (i)

m₁ (v₁_i - v₁_f) = m₂ (v₁_i - v₂_i + v₁_f - v₂_i)

or m₁v₁_i - m₁v₁_f = m₂v₁_i - 2m₂v₂_i + m₂v₁_f

or -m₁v₁_f - m₂v₁_f = -m₁v₁_i + m₂v₁_i - 2m₂v₂_i

or (m₁ + m₂) v₁_f = (m₁ - m₂) v₁_i + 2m₂v₂_i

Again, from equation (iii),

v₁_f = v₂_f - v₁_i + v₂_i

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

Sub Topics

Special cases

Special cases

Case (i) when m₁ = m₂_, 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 v₂_i = 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 = m₂_, i.e., when the colliding bodies are of the same mass.

From equation (vi)

v₁_f = 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.