CH 4 Notes: Momentum and Impulse

Key Concepts

• Momentum: $p = m v$

• Impulse: $\mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} \, dt$

• For constant force: $\mathbf{J} = \mathbf{F} \Delta t$

• Impulse-momentum theorem: $\Delta \mathbf{p} = \mathbf{J}$

• Units: momentum in kg·m/s, impulse in N·s (since $1\,\text{N} = 1\,\text{kg}\cdot\text{m}/\text{s}^2$, so N·s = kg·m/s)

• Direction: impulse and momentum change align with the direction of the applied force

• Core idea: the effect of a force on motion is determined by the impulse (not just the peak force)

• Related principles: longer contact time or larger force increases impulse; momentum is conserved in absence of external impulse

4.1 What happens to momentum depends on the impulse

• Statement from transcript: “Everything depends on the impulse” (paraphrase of impulse-dominance in changing motion)

• Momentum of an object changes by the impulse applied: $\Delta \mathbf{p} = \mathbf{J} = \int \mathbf{F} dt$

• If a force acts over a time interval, the resulting change in momentum equals the impulse delivered during that interval

• Real-world intuition:

  • A short, sharp push delivers a large instantaneous force but may have a smaller or larger impulse depending on duration; what matters for momentum change is the product of force and time, not just the peak force

  • A longer push with a smaller peak force can produce a larger impulse if the duration is longer

• Example framing for exam: If a force acts on a body for time Δt, the momentum change is $\Delta p = F \Delta t$ (for constant F) or more generally $\Delta p = \int F dt$

• Significance: impulse bridges force and momentum; it explains why identical forces applied over longer times impart more momentum

4.2 How impulse differs from force

• Force is a push or pull at a given instant (or as a function of time): a vector quantity, can vary with time

• Impulse is the cumulative effect of force over a period of time; it is not a force but the result of applying a force over time

• Mathematical relationships:

  • $\mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} \, dt$

  • For constant force: $\mathbf{J} = \mathbf{F} \Delta t$

• Key differences:

  • Force describes instantaneous interaction; impulse describes the total change in motion due to that interaction

  • Impulse depends on both magnitude of force and duration; two different force-time profiles can yield the same impulse and same change in momentum

• Practical interpretation:

  • A very brief hit (large peak force, very short time) may impart the same momentum as a gentler hit over a longer time if the impulses match

• Core takeaway: impulse is the cause (through time) of momentum change; force is the cause of impulse but not itself the momentum change

4.3 For the same force, which cannon imparts the greater speed to a cannonball? (long cannon vs short cannon)

• Given: same propulsion force F acts on the cannonball inside the barrel during launch

• Key idea: impulse depends on time the force acts: $\mathbf{J} = F \Delta t$ for constant F

• Longer barrel implies longer contact time (ballooning out the acceleration distance) or a longer acceleration path, so the time during which the force acts, Δt, is larger in the longer cannon

• Consequence:

  • Longer cannon → larger impulse: J{long} = F \Delta t{long} > F \Delta t{short} = J{short}

  • Larger impulse → greater change in momentum: $\Delta p = J$

  • If initial velocity is zero, final velocity after propulsion is $v_f = \dfrac{\Delta p}{m} = \dfrac{J}{m}$

• Therefore, the long cannon imparts greater exit speed to the cannonball (assuming the same force profile and mass of the cannonball)

• Summary equation tying it together:

  • If initial velocity is zero: $v_f = \dfrac{J}{m} = \dfrac{F \Delta t}{m}$ for constant F

• Practical nuance: real cannons involve varying force during ignition and gas expansion; the simplified model uses impulse to explain why longer barrels can yield higher muzzle velocity under the same propulsion force

Connections to fundamentals and real-world relevance

• Momentum conservation: in absence of external impulse, total momentum is conserved; impulse is the external change to momentum

• In sports and safety: athletes optimize impulse (e.g., baseball batting, football kicking) to maximize momentum transfer; protective gear often reduces impulse by increasing contact time or spreading force over longer durations

• Engineering implications: designing devices that apply force over time to achieve desired momentum changes (e.g., braking systems, airbags)

• Ethical/practical note: understanding impulse helps assess safety and risk in collisions and impacts

Quick recap formulas to memorize

• Momentum: $p = m v$

• Impulse: $\mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} \, dt$

• Constant force impulse: $\mathbf{J} = \mathbf{F} \Delta t$

• Relationship to velocity (when initial velocity is 0): $v_f = \dfrac{J}{m} = \dfrac{F \Delta t}{m}$

• Units overview:

  • momentum: kg·m/s

  • impulse: N·s (kg·m/s)