Understanding the Concepts

* Kinetic Energy: The energy possessed by an object due to its motion. It's calculated as KE = (1/2)mv², where m is mass and v is velocity.

* Linear Momentum: A measure of an object's mass in motion. It's calculated as p = mv, where m is mass and v is velocity.

Solving the Problem

1. Relating Momentum and Velocity: Since the two particles have equal linear momentum (p), we can write:

p₁ = p₂

m₁v₁ = m₂v₂

v₂ = (m₁/m₂)v₁

2. Finding the Ratio of Kinetic Energies: Let's denote the kinetic energy of the 1g particle as KE₁ and the kinetic energy of the 4g particle as KE₂.

KE₁ = (1/2)m₁v₁²

KE₂ = (1/2)m₂v₂²

Substitute v₂ from step 1:

KE₂ = (1/2)m₂[(m₁/m₂)v₁]²

KE₂ = (1/2)(m₁²/m₂)v₁²

Now, find the ratio KE₁/KE₂:

KE₁/KE₂ = [(1/2)m₁v₁²] / [(1/2)(m₁²/m₂)v₁²]

KE₁/KE₂ = m₂/m₁

3. Substituting the Masses: We know m₁ = 1g and m₂ = 4g.

KE₁/KE₂ = 4g / 1g = 4

Answer: The ratio of kinetic energies between the two particles is 4:1. This means the particle with a mass of 4g has four times the kinetic energy of the particle with a mass of 1g, even though they have equal linear momentum.