Question

Starting with equations (m₁v₁ = m₁v'₁costheta₁ + m₂v'₂costheta₂) and (0 = m₁v'₁sintheta₁ + m₂v'₂sintheta₂) for conservation of momentum in the x- and y-directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses:

a) 12mv₁² = 12m(v'₁)² + 12m(v'₂)² + mv'₁v'₂cos(θ₁ - θ₂)
b) 14mv₁² = 14m(v'₁)² + 14m(v'₂)² + 12mv'₁v'₂cos(θ₁ - θ₂)
c) 12mv₁² = 12m(v'₁)² + 12m(v'₂)² - mv'₁v'₂cos(θ₁ - θ₂)
d) 14mv₁² = 14m(v'₁)² + 14m(v'₂)² - 12mv'₁v'₂cos(θ₁ - θ₂)

Answer 1

The equation for the conservation of internal kinetic energy in an elastic collision of two objects with equal masses is 12mv₁² = 12m(v'₁)² + 12m(v'₂)² + mv'₁v'₂cos(θ₁ - θ₂).

Step-by-step explanation:

For an elastic collision of two objects of equal masses, the equation for conservation of internal kinetic energy is 12mv₁² = 12m(v'₁)² + 12m(v'₂)² + mv'₁v'₂cos(θ₁ - θ₂).

This equation represents the conservation of momentum in the x- and y-directions, where one object is initially stationary. It shows that the initial kinetic energy of the system is equal to the final kinetic energy of the two objects.

The equation is derived from the principle of conservation of momentum, where both momentum and kinetic energy are conserved in an elastic collision.