Question

An object of mass M and moving at velocity V collides with a stationary object of mass 3M. The collision is elastic. After the collision the object with mass M has a final velocity of v₁, and the object with mass 3M has a final velocity v₂. Show that the relationship between the final velocities v₁, and v₂ is v₁ = -v₂.

Answer 1

The final velocities v1 and v2 after an elastic collision between two objects where one is stationary can be related using conservation of momentum and conservation of kinetic energy principles, showing that v1 equals -v2.

Step-by-step explanation:

To demonstrate the relationship between the final velocities v1 and v2 after an elastic collision between an object of mass M and one with mass 3M, we will use two fundamental principles: conservation of momentum and conservation of kinetic energy.

Momentum before the collision is MV, as the larger mass is stationary. After the collision, the momentum is Mv1 + 3Mv2. By conservation of momentum, MV = Mv1 + 3Mv2. As the collision is elastic, kinetic energy is also conserved. The initial kinetic energy is (1/2)MV2 and the final kinetic energy is (1/2)Mv12 + (1/2)(3M)v22. Equating the initial and final kinetic energies and using the equation for conservation of momentum, we can solve for the velocities to find v1 = -v2.