Question

Derive the equations giving the final speeds for two objects that collide elastically, with the mass of the objects being m₁ and m₂ and the initial speeds being v₁,ᵢ and v₂,ᵢ = 0 (i.e., second object is initially stationary).

a) (v₁, = {m₁ - m₂}/{m₁ + m₂} v₁,ᵢ), (v₂, = {2m₁}/{m₁ + m₂} v₁,ᵢ)
b) (v₁, = {m₁}/{m₁ + m₂} v₁,ᵢ), (v₂, = {2m₂}/{m₁ + m₂} v₁,ᵢ)
c) (v₁, = {m₁ + m₂}/{m₁ - m₂} v₁,ᵢ), (v₂, = {m₁ - m₂}/{m₁ + m₂} v₁,ᵢ)
d) (v₁, = v₁,ᵢ), (v₂, = 0)

Answer 1

Using conservation of momentum and energy, the final velocities of two objects in an elastic collision are derived as v'₁ = {(m₁ - m₂)/(m₁ + m₂)} v₁,ᵢ and v'₂ = {2m₁/(m₁ + m₂)} v₁,ᵢ, when the second object is initially stationary. a) (v₁, = {m₁ - m₂}/{m₁ + m₂} v₁,ᵢ), (v₂, = {2m₁}/{m₁ + m₂} v₁,ᵢ) is correct option.

Step-by-step explanation:

To derive the final speeds for two objects that collide elastically with masses m₁ and m₂, and initial speeds v₁,ᵢ and v₂,ᵢ = 0 (with the second object initially stationary), we can use the conservation of momentum and conservation of kinetic energy principles.

Since momentum is conserved, we have m₁v₁,ᵢ = m₁v'₁ + m₂v'₂. Because the collision is elastic, kinetic energy is also conserved, which gives us ½m₁v₁,ᵢ² = ½m₁v'₁² + ½m₂v'₂².

Solving these equations, we find the final velocities are: v'₁ = {(m₁ - m₂)/(m₁ + m₂)} v₁,ᵢ (Answer A) and v'₂ = {2m₁/(m₁ + m₂)} v₁,ᵢ. These equations represent the final velocities of m₁ and m₂ respectively after the collision.