--- title: "Principle of impulse and momentum" subject: "Engineering Mechanics" chapter: 4 key: "semester_3/engineering_mechanics:impulse-momentum-principle-particle" source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/impulse-momentum-principle-particle site: BCT Engineering --- # Principle of impulse and momentum Engineering Mechanics · Chapter 4: Kinetics of Particles: Impulse and Momentum - Asked in: year not given, 5 marks - Video: [Linear Impulse and Momentum (learn to solve any problem)](https://www.youtube.com/watch?v=bl2gEps8pEk) — Question Solutions - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/impulse-momentum-principle-particle - Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-4 ## Question State and derive the principle of impulse and momentum for a particle. Explain impulsive force and show that linear momentum of a system of particles is conserved when no external impulse acts. ## Short answer The impulse of the resultant force equals the change in linear momentum: integral F dt = m v₂ – m v₁, derived by integrating Newton's second law. Impulsive forces are large and brief; non-impulsive forces are neglected during impact. For a system with no external impulse, total linear momentum remains constant. ## Answer #### Principle **Linear impulse** of a force is $\int \mathbf{F}\,dt$ (unit N·s). **Linear momentum** is $m\mathbf{v}$ (unit kg·m/s). The principle states that the linear impulse of the resultant force acting on a particle during a time interval equals the change in its linear momentum: $$ m\mathbf{v}_1 + \int_{t_1}^{t_2} \mathbf{F}\,dt = m\mathbf{v}_2 $$ #### Derivation Newton's second law: $\mathbf{F} = m\dfrac{d\mathbf{v}}{dt}$ (constant mass). Multiply by $dt$ and integrate from $t_1$ to $t_2$: $$ \int_{t_1}^{t_2}\mathbf{F}\,dt = m\int_{\mathbf{v}_1}^{\mathbf{v}_2} d\mathbf{v} = m\mathbf{v}_2 - m\mathbf{v}_1 $$ It is a vector equation, so it is applied separately in each direction. It is useful when forces depend on time and when time (not distance) is required. #### Impulsive force An **impulsive force** is a very large force acting for a very short time (for example in a collision or a hammer blow), so that its impulse is finite while the displacement during the action is negligible. Non-impulsive forces such as weight and spring forces are ignored during the impact. #### Conservation of linear momentum For a system of particles, the internal forces occur in equal and opposite pairs (Newton's third law), so their impulses cancel. If the external impulse in a direction is zero, the total momentum of the system in that direction is the same before and after: $$ \sum m_i \mathbf{u}_i = \sum m_i \mathbf{v}_i $$ Examples are an explosion, a recoiling gun, and a collision between two bodies. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The principle of impulse and momentum states that: A. Impulse equals the change in kinetic energy B. Linear impulse of the resultant force equals the change in linear momentum C. Momentum equals force times distance D. Impulse equals force divided by time Answer: B. m v1 + integral of F dt = m v2. (Key point: Principle) 2. How is the principle derived? A. Integrate F = m dv/dt after multiplying by dt B. Integrate F ds = m v dv C. Differentiate momentum twice D. Use conservation of energy Answer: A. Multiplying F = m dv/dt by dt and integrating gives integral F dt = m v2 - m v1. (Key point: Derivation) 3. Which describes an impulsive force? A. A large force acting a very short time with finite impulse and negligible displacement B. A small force acting a long time C. Any constant force D. A force that does no work Answer: A. In impacts, weight and spring forces are non-impulsive and ignored during the impact. (Key point: Impulsive force) 4. Why is the momentum of a system conserved when there is no external impulse? A. Internal forces are zero B. Internal forces occur in equal and opposite pairs whose impulses cancel C. Energy is conserved D. Velocities do not change Answer: B. By Newton's third law the internal impulses cancel, so total momentum is unchanged. 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