Collisions - Expressions for Final Velocities in Elastic Head-On Collision

In a perfectly elastic collision in one dimension (head‑on), both the total linear momentum and the total kinetic energy of the two‐body system are conserved. This allows us to derive formulas for the final velocities of each mass after the collision.

Symbols used:

• m₁, m₂ = masses of body 1 and body 2

• u₁, u₂​ = initial velocities (before collision) of masses 1 & 2

• v₁, v₂​ = final velocities (after collision) of masses 1 & 2

1. Conservation of Linear Momentum:
   m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
2. Conservation of Kinetic Energy:
   \[\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2\]

1. From energy conservation, we can show that
   u₁ + v₁ = v₂ + u₂​
   (i.e., the relative speed of separation equals the relative speed of approach).
2. Rearranging and using the momentum equation, we substitute and simplify (steps omitted here for brevity; helpful to work out line‑by‑line).
3. The result:
   v₁ = \[\frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2\]                                                      ---(1)
   v₂ = \[\frac{m_2-m_1}{m_1+m_2}u_2+\frac{2m_1}{m_1+m_2}u_1\]   ---(2)

1. Equal Masses (m₁ = m₂):

   v₁ = u₂, v₂ = u₁

   The bodies exchange velocities.

2. Heavier Striker (m₁ ≫ m₂, u₂ = 0):

   v₁ ≈ u₁, v₂ ≈ 2u₁

   Lighter body rebounds with twice the speed.

3. Heavier Target (m₁ ≪ m₂, u₂ = 0):

   v₁ ≈ −u₁, v₂ ≈ 0

   Lighter body rebounds back, heavier body remains almost stationary.

In a partially elastic collision (not perfectly elastic), we use the coefficient of restitution eee defined by
e = \[(\frac{speed\ of\ separation}{speed\ of\ approach}) = \frac{v_{2}-v_{1}}{u_{1}-u_{2}}\]​​

Using momentum conservation together with this definition, one can derive for equal masses (e.g., two identical marbles with one initially at rest) that
​\[\frac{v_2}{v_1}=\frac{1+e}{1-e}\]

Hence, the ratio of final speeds depends directly on e.
e = \[\frac{v_2-v_1}{u_1-u_2}=\frac{v_2-v_1}{u_1}\]

Using momentum conservation:

v₁ + v₂ = u₁ ⇒ \[\frac{v_2}{v_1}=\frac{1+e}{1-e}\]

• When a rubber ball hits the ground, the Earth acts as a “massive” second body. The ball rebounds nearly with the same speed—it’s an example of an elastic collision in everyday life.
• When two identical billiard balls collide head‑on, the moving ball stops and the stationary one takes off: a direct illustration of the equal‑mass case.
• Consider a heavy truck hitting a light car (in an idealised “elastic” sense): the truck is hardly affected, the car may fly off fast → heavy–light mass scenario.