Change in Momentum and Impulse
The most general Newton’s-second-law form in this unit is F⃗_ext=dP⃗_sys/dt. Integrating that relation over time gives impulse J⃗=∫F⃗_ext dt=ΔP⃗. The same relationship appears graphically as area under a force-time curve and slope on a momentum-time graph, so calculus, vectors and experimental data all describe the same momentum transfer.
Topic sequence · 2 of 4
Learning objectives
Objective 1
Describe impulse as the time integral of net external force and relate it to momentum change.
Objective 2
Translate among F(t) area, P(t) slope, vector momentum change and average-force representations.
Objective 3
Use the impulse-momentum theorem experimentally and explain the constant-mass and changing-mass scope in the current course.
- Topic 4.1 momentum and system momentum.
- Topic 2.5 Newton’s second law.
- Topic 1.2 derivatives and definite integrals.
- Topic 1.3 graph slope and signed-area interpretations.
Definitions first
Representations you must read and create
Representation 1
Force-versus-time graph whose signed area equals the impulse in the chosen component.
Representation 2
Momentum-versus-time graph whose tangent slope equals net external force.
Representation 3
Impulse vector drawn head-to-tail with initial momentum to produce final momentum: P⃗_f=P⃗_i+J⃗.
Representation 4
Force-sensor and motion-sensor data table for experimental comparison of ∫Fdt and Δp.
Core relationships and calculus
Variables: F⃗_ext net external force (N); P⃗_sys system momentum (kg·m/s)
Use: momentum form of Newton’s second law
Variables: J⃗ impulse (N·s)
Use: impulse from a variable force
Variables: all vector quantities
Use: impulse-momentum theorem
Variables: F⃗_avg average net force over Δt
Use: replace a varying force by an impulse-equivalent average
Variables: component-wise derivative
Use: read force from momentum-versus-time data
Start from F⃗_ext=dP⃗/dt and integrate over the interaction interval: ∫F⃗_ext dt=∫dP⃗=P⃗_f−P⃗_i. This is the impulse-momentum theorem. For constant mass, d(mv⃗)/dt=m a⃗ and the familiar F⃗=ma⃗ follows. The current framework also asks students to recognize a changing-mass case in which velocity is constant while momentum changes.
Integrate Newton’s momentum form from t_i to t_f: ∫F⃗_ext dt=∫(dP⃗/dt)dt=P⃗_f−P⃗_i. For constant mass, P⃗=m v⃗ gives F⃗_ext=m d v⃗/dt=m a⃗. If the course presents the special case of constant velocity with changing mass, dP⃗/dt=v⃗ dm/dt, showing why the momentum form is broader than simply writing ma.
The current framework includes the special conceptual case in which velocity is constant while mass changes, giving F⃗_ext=dP⃗/dt=v⃗ dm/dt. Full rocket-equation derivations and arbitrary variable-mass-system dynamics require careful system definitions and are beyond what this lesson treats quantitatively.
Physical examples
Example 1
A large force acting briefly and a smaller force acting longer can deliver the same impulse if the signed areas under their force-time curves are equal.
Example 2
A padded collision increases the stopping time for approximately the same change in momentum, reducing the average force magnitude.
Example 3
If a force-time graph crosses the axis, positive and negative areas must be combined with sign; geometric area alone is not the impulse.
Example 4
A straight-line momentum-time graph has constant slope and therefore constant net external force.
Example 5
During a very short bat–ball collision, gravity acts but its impulse can be tiny compared with the contact-force impulse. Neglecting gravity over the impact interval is an impulse approximation, not a claim that gravity vanishes.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
On F(t), calculate signed area. On P(t), calculate slope. Do not interchange the two graph operations.
Reasoning pattern 2
If the question gives the same Δp but a longer interaction time, the average force magnitude is smaller because F_avg=Δp/Δt.
Reasoning pattern 3
Choose the system before labeling a force “external.” Internal third-law force pairs do not change total system momentum.
Reasoning pattern 4
For a variable force given analytically, integrate with physical time limits and keep vector/component signs through the calculation.
Reasoning pattern 5
When using an impulse approximation, compare impulse scales or interaction times; do not simply delete a force because the collision is brief.
Deeper calculus and limiting-case check
Impulse depends on the time integral of force, not on the peak force alone. Two force pulses with very different shapes can produce the same final momentum if their signed areas are equal. This makes impulse a useful bridge between detailed collision dynamics and the state-to-state conservation analysis of Topic 4.3.
Model-selection workflow: decide what the problem is before choosing an equation
For Change in Momentum and Impulse, the fastest reliable route is not equation hunting. Start by identifying the system and time interval, encode it with the momentum/impulse representation, choose the impulse-momentum or conservation equation, and then test the external-impulse assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat impulse and impulse-momentum theorem as physical quantities first and symbols second.
Choose a representation
Build Force-versus-time graph whose signed area equals the impulse in the chosen component.. If the prompt supplies Momentum-versus-time graph whose tangent slope equals net external force., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is F⃗_ext = dP⃗_sys/dt. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current framework includes the special conceptual case in which velocity is constant while mass changes, giving F⃗_ext=dP⃗/dt=v⃗ dm/dt. Full rocket-equation derivations and arbitrary variable-mass-system dynamics require careful system definitions and are beyond what this lesson treats quantitatively. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Force-versus-time graph whose signed area equals the impulse in the chosen component. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | F⃗_ext = dP⃗_sys/dt | momentum form of Newton’s second law. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | J⃗_ext = ∫_{t_i}^{t_f} F⃗_ext(t) dt | impulse from a variable force. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Start from F⃗_ext=dP⃗/dt and integrate over the interaction interval: ∫F⃗_ext dt=∫dP⃗=P⃗_f−P⃗_i. This is the impulse-momentum theorem. For constant mass, d(mv⃗)/dt=m a⃗ and the familiar F⃗=ma⃗ follows. The current framework also asks students to recognize a changing-mass case in which velocity is constant while momentum changes. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A net force varies as F_x(t)=400−200t N from t=0 to 2.0 s. The impulse is J_x=∫₀²(400−200t)dt=[400t−100t²]₀²=400 N·s. If the object initially has p_x=−100 kg·m/s, then p_{x,f}=300 kg·m/s. The force reaches zero at 2.0 s, but the accumulated impulse is positive because the force was positive throughout the interval. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 4.2
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship F⃗_ext = dP⃗_sys/dt should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Force-versus-time graph whose signed area equals the impulse in the chosen component. and Momentum-versus-time graph whose tangent slope equals net external force.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a force sensor and motion sensor on a cart. Record F(t) during a collision, integrate the force-time data numerically to obtain impulse, independently calculate m(v_f−v_i), and compare the two values with uncertainty. Then compare peak force and impulse for different padding durations.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using peak force times the entire collision duration when the force is not constant; integrate the actual curve or use a justified average force. | Return to the momentum/impulse representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Ignoring sign on force-time area or momentum components. | Return to the impulse-momentum or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using an internal interaction force as though it were the net external force on the whole selected system. | Return to the momentum/impulse representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming F=ma is always the most general statement; F=dP/dt is the system-level momentum form used here. | Return to the impulse-momentum or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.5 supplies Newton’s second law. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 1.3 supplies slope/area graph reasoning. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 4.1 defines P⃗ so its derivative and change have physical meaning. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write F⃗_ext = dP⃗_sys/dt and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 4.2 mastery ladder
Interactive force-time impulse integral
For F(t)=at+b in one component, integrate from t₁ to t₂ and apply J=Δp.
Experimental connection
Use a force sensor and motion sensor on a cart. Record F(t) during a collision, integrate the force-time data numerically to obtain impulse, independently calculate m(v_f−v_i), and compare the two values with uncertainty. Then compare peak force and impulse for different padding durations.
Connections across Physics C
Connection 1
Topic 2.5 supplies Newton’s second law.
Connection 2
Topic 1.3 supplies slope/area graph reasoning.
Connection 3
Topic 4.1 defines P⃗ so its derivative and change have physical meaning.
Connection 4
Topic 4.3 uses the fact that zero external impulse means no change in total system momentum.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using peak force times the entire collision duration when the force is not constant; integrate the actual curve or use a justified average force.
Avoid this
Ignoring sign on force-time area or momentum components.
Avoid this
Using an internal interaction force as though it were the net external force on the whole selected system.
Avoid this
Assuming F=ma is always the most general statement; F=dP/dt is the system-level momentum form used here.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Is impulse a force?
No. Impulse is force accumulated over time and has units N·s, the same dimensions as momentum.
What does area under a force-time graph mean?
The signed area equals the impulse in that force component, which equals the corresponding momentum change of the selected system.
What does the slope of a momentum-time graph mean?
The slope dP/dt is the net external force on the selected system.
Why can padding reduce collision force?
For approximately the same Δp, increasing the interaction time reduces the required average force magnitude.