Class 11 > Unit # 03: Dynamics > Elastic Collision in One Dimension

Talha's Physics Academy

Unit # 3: Dynamics — Elastic & Inelastic Collisions

Lecture Overview

This lecture from Talha's Physics Academy covers the definitions of elastic and inelastic collisions, along with the complete step-by-step mathematical derivation for the final velocities of two bodies undergoing a one-dimensional elastic collision (Class 11 Physics, Unit 3: Dynamics).

Source Video: Derivation of final velocities for Elastic Collision in one Dimension - Unit 3 Dynamics - #TP11 30

1. Definitions of Collision Types

Elastic Collision

It is a collision in which both momentum and total kinetic energy of the system before and after the collision are conserved.

Inelastic Collision

It is a collision in which the total momentum of the system remains conserved, but the total kinetic energy before and after the collision changes (is not conserved).

2. Derivation for Final Velocities in One-Dimensional Elastic Collision

Consider two non-rotating spheres of masses \(m_1\) and \(m_2\) moving initially with velocities \(u_1\) and \(u_2\) along a straight line in the same direction (with \(u_1 > u_2\)). After an elastic collision in one dimension, their final velocities become \(v_1\) and \(v_2\).

Step 1: Conservation of Momentum

According to the law of conservation of momentum:

$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$

Rearranging terms to group masses together:

$$m_1(u_1 - v_1) = m_2(v_2 - u_2) \quad \text{--- (1)}$$

Step 2: Conservation of Kinetic Energy

For an elastic collision, total kinetic energy is conserved:

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

Multiplying by 2 and rearranging terms:

$$m_1(u_1^2 - v_1^2) = m_2(v_2^2 - u_2^2)$$

Applying the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\):

$$m_1(u_1 - v_1)(u_1 + v_1) = m_2(v_2 - u_2)(v_2 + u_2) \quad \text{--- (2)}$$

Step 3: Relative Velocity Equation

Dividing equation (2) by equation (1):

$$u_1 + v_1 = v_2 + u_2 \quad \text{--- (3)}$$

Step 4: Final Velocity of the First Body (\(v_1\))

From equation (3), express \(v_2 = u_1 + v_1 - u_2\) and substitute it into equation (1). Simplifying yields:

$$v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2} \quad \text{--- (A)}$$

Step 5: Final Velocity of the Second Body (\(v_2\))

Similarly, expressing \(v_1 = v_2 + u_2 - u_1\) from equation (3) and substituting into equation (1) yields:

$$v_2 = \frac{2m_1u_1 + (m_1 - m_2)u_2}{m_1 + m_2} \quad \text{--- (B)}$$

© 2026 Talha's Physics Academy. All rights reserved.