One-Dimensional Elastic Collision of Two Carts

Problem Statement

Cart A of mass 0.8 kg moves to the right at 4 m/s on a smooth track. Cart B of mass 1.2 kg moves toward it at 2 m/s (to the left). The carts collide head-on and the collision is perfectly elastic. Determine the final velocities of both carts and check that kinetic energy is conserved.

Given / Required

Given: masses m_A = 0.8 kg, m_B = 1.2 kg. Initial velocities u_A = +4 m/s, u_B = -2 m/s.

Required: final velocities v_A and v_B, plus verification of kinetic energy conservation.

Hint

Set up equations for conservation of linear momentum and conservation of kinetic energy. Solve the two simultaneous equations or use the relative velocity shortcut for elastic collisions.

Step-by-Step Solution

Step 1 – Conservation of Momentum

\[ m_A u_A + m_B u_B = m_A v_A + m_B v_B \]

Substitute values:

\[ 0.8(4) + 1.2(-2) = 0.8 v_A + 1.2 v_B \]

\[ 3.2 - 2.4 = 0.8 v_A + 1.2 v_B \]

\[ 0.8 = 0.8 v_A + 1.2 v_B \]

Step 2 – Relative Velocity Relation

For elastic collisions: \(u_A - u_B = -(v_A - v_B)\).

\[ 4 - (-2) = -(v_A - v_B) \]

\[ 6 = -(v_A - v_B) \]

\[ v_A - v_B = -6 \]

Step 3 – Solve Simultaneous Equations

From momentum: \(0.8 = 0.8 v_A + 1.2 v_B\).

Use v_A = v_B - 6.

\[ 0.8 = 0.8(v_B - 6) + 1.2 v_B = 0.8 v_B - 4.8 + 1.2 v_B = 2.0 v_B - 4.8 \]

\[ 2.0 v_B = 5.6 \implies v_B = 2.8,\text{m/s} \]

Then \(v_A = v_B - 6 = -3.2,\text{m/s}\).

Step 4 – Check Kinetic Energy

Initial kinetic energy:

\[ K_i = \tfrac{1}{2} (0.8)(4^2) + \tfrac{1}{2} (1.2)(2^2) = 6.4 + 2.4 = 8.8,\text{J} \]

Final kinetic energy:

\[ K_f = \tfrac{1}{2} (0.8)(-3.2)^2 + \tfrac{1}{2} (1.2)(2.8)^2 = 4.096 + 4.704 \approx 8.80,\text{J} \]

Energies match, so the solution is consistent.

Final Answer

• Cart A rebounds with v_A = -3.2 m/s.
• Cart B moves forward with v_B = +2.8 m/s.
• Total kinetic energy is conserved (8.8 J).

Extension / Variation

• Change one mass or add slight inelasticity and see how speeds adjust.
• Explore the special case where masses are equal.

Key Concept Recap

• Momentum and kinetic energy are both conserved in elastic collisions.
• Relative speed of approach equals relative speed of separation.
• Signs matter; choose a clear positive direction before writing equations.