CH 4 Notes: Momentum and Impulse

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

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

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

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

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

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)