A force F₁ acts on a particle, accelerating it from rest to a velocity v . Subsequently, F₁ is replaced by a force F₂ that decelerates the particle back to rest.

Step-by-step solution

Let the mass of the particle be m . During the acceleration phase, the particle starts from rest and attains a velocity v . Let t₁ be the time for which the force F₁ acts. Using the impulse-momentum theorem, we have: F₁ t₁ = m v - 0 = m v During the deceleration phase, the particle is brought back to rest from velocity v . Let t₂ be the time for which the force F₂ acts. The magnitude of the change in momentum is the same. Thus, we have: F₂ t₂ = m v - 0 = m v Equating the two expressions, we get: F₁ t₁ = F₂ t₂ Since