Question

Explain elastic collision in two dimensions.

Answer 1

Elastic collision in two dimensions :- Consider a particle of mass m₁ moving along x-axis with initial velocity u₁ collides with another particle of mass m₂ at rest. After, collision, let the mass m₁ moves with velocity v₁ at an angle θ₁ w.r.t. the original path (i.e., x-axis) and particle m₂ moves with velocity v₂ at an angle θ₂ with respect to the horizontal direction (i.e., x-axis).

According to the law of conservation of linear momentum

m₁ vector n₁ + 0 = m₁ vector v₁ + m₂ vector v₂

or, m₁ vector u₁ = m₁ vector v₁ + m₂ vector v₂   ...(i)

The x-component of equation (i) gives:

m₁u₁ = m₁v₁ cosθ₁ + m₂v₂ cosθ₂    ...(ii)

and y-component of equation (i) gives

0 = m₁v₁ sinθ₁ - m₂v₂ sinθ₂   ...(iii)

As the collision is elastic, so kinetic energy is also conserved.

i.e., 1/2 m₁u₁² = 1/2 m₁v₁² + 1/2 m₂v₂²   ...(iv)

Assume that m₁, m₂ and u₁, are known. Now, the motion after collision involves four unknown quantities, i.e., v₁, v₂, θ₁ and θ₂. To evaluate these four quantities, we need four equations. But, we have only three equations (ii) to (iv). So, we cannot determine these four quantities. So, let θ₁ be also known, then v₁, v₂ and θ₂ and can be calculated.