Two particles having masses m₁ and m₂ undergo a head-on collision. Their initial speeds are u₁ and u₂ , travelling in the same direction. The collision commences at t = 0 and the particles interact over a time interval t . Throughout the collision, the speed of the first particle changes according to the relation v(t) = u₁ + t t (v₁ - u₁) . Determine the speed of the second particle as a function of time while the co

Step-by-step solution

By the principle of conservation of linear momentum, the total momentum of the system remains constant throughout the collision. m₁ u₁ + m₂ u₂ = m₁ v₁(t) + m₂ v₂(t) Given the velocity of the first particle as a function of time: v₁(t) = u₁ + t t (v₁ - u₁) Substituting v₁(t) into the momentum conservation equation: m₁ u₁ + m₂ u₂ = m₁ ( u₁ + t t (v₁ - u₁) ) + m₂ v₂(t) m₁ u₁ + m₂ u₂ = m₁ u₁ + m₁ t t (v₁ - u₁) + m₂ v₂(t) Cancelling m₁ u₁ from both sides: m₂ u₂ = m₁ t t (v₁ - u₁) + m₂ v₂(t) Rearranging to solve for v₂(t