State and prove the impulse-momentum theorem

Impulse–Momentum Theorem

The impulse–momentum theorem states that the impulse acting on a body is equal to the change in momentum of the body.

J = Δp

where:

J = impulse

Δp = change in momentum = m v − m u

Proof

1. From Newton’s second law: F = dp/dt
2. Multiply both sides by dt: F · dt = dp
3. Integrate over time t₁ to t₂: ∫_t₁^t₂ F dt = ∫_pᵢ^pᶠ dp
4. The left side is impulse (J), the right side is change in momentum: J = pᶠ − pᵢ

Therefore:

J = m v − m u