Momentum and Impulse

Introduction

Momentum and impulse give you a powerful way to describe what happens when forces act over time, especially during collisions. In this aiMOOC for Grades 11–13, you will connect Newton's laws, vectors, linear momentum, impulse, collisions, and conservation of momentum. You will calculate with equations, interpret force–time graphs, analyze one-dimensional and two-dimensional interactions, and explain safety technologies such as helmets, seat belts, airbags, and crumple zones.

The central idea is simple but deep: a net external force changes momentum. When you look at a force over a time interval, the accumulated effect is impulse. For a system whose net external impulse is negligible, total momentum remains constant even though the individual objects may exchange momentum.

Datei:Impuls Masse Geschwindigkeit.svg

The image emphasizes that momentum depends on both mass and velocity. Because velocity is a vector, momentum is also a vector.

The review video above is useful after your first reading of the course. As you watch, pause whenever a sign convention or vector direction appears and explain why the sign matters.

Learning Goals

By the end of this aiMOOC, you should be able to explain momentum and impulse conceptually, use their SI units correctly, apply the impulse–momentum theorem, identify when total system momentum is conserved, distinguish elastic from inelastic collisions, solve one-dimensional and component-based two-dimensional collision problems, interpret areas under force–time graphs, and evaluate real-world strategies that reduce impact force.

For advanced work, you should also be able to connect Newton's second law in momentum form, ${\vec{F}}_{n e t} = d \vec{p} / d t$ , with the integral definition of impulse and discuss the assumptions behind idealized collision models.

Linear Momentum

Definition and Vector Nature

For an object with constant mass, linear momentum is

$\vec{p} = m \vec{v}$

where $\vec{p}$ is momentum, $m$ is mass, and $\vec{v}$ is velocity. The SI unit is kilogram metre per second, written $k g m s^{- 1}$ . Momentum points in the same direction as velocity because mass is a positive scalar.

A 1200 kg car moving east at 15 m/s has momentum of 18,000 kg m/s east. If east is defined as positive, its momentum is +18,000 kg m/s. The same car moving west at the same speed has momentum −18,000 kg m/s in a one-dimensional sign convention. A negative momentum value does not mean "less momentum"; it encodes the chosen direction.

Momentum is additive. For a system of several objects,

$\vec{P} = \sum_{i} {\vec{p}}_{i}$ .

This means you must add momenta as vectors. In one dimension, signed values are sufficient. In two dimensions, add x-components and y-components separately.

Momentum and Newton's Second Law

Newton's second law can be written more generally as

${\vec{F}}_{n e t} = \frac{d \vec{p}}{d t}$ .

If mass is constant, then $d \vec{p} / d t = m d \vec{v} / d t = m \vec{a}$ , which gives the familiar form ${\vec{F}}_{n e t} = m \vec{a}$ . The momentum form is especially useful because it highlights that net external force changes momentum.

Do not confuse momentum with kinetic energy. Momentum is a vector and depends linearly on velocity. Translational kinetic energy, $K = \frac{1}{2} m v^{2}$ , is a scalar and depends on the square of speed. Two objects can have the same momentum magnitude but different kinetic energies.

Impulse

Force Acting Over Time

Impulse describes the accumulated effect of a force over a time interval. The net impulse is

$\vec{J} = \int_{t_{i}}^{t_{f}} {\vec{F}}_{n e t} (t) d t$ .

For a constant net force, or when an average force is used,

$\vec{J} = {\vec{F}}_{a v g} Δ t$ .

Impulse has SI units of newton seconds, $N s$ . Because one newton equals one kilogram metre per second squared, $N s$ is equivalent to $k g m s^{- 1}$ . Impulse and momentum therefore have the same dimensions and units.

Datei:Average-force-and-impulse.svg

The diagram above compares a changing force with an equivalent constant average force that produces the same impulse. The important comparison is the total force–time area, not whether the force is constant at every instant.

The video develops impulse as change in momentum and connects contact time to impact force.

Area Under a Force-Time Graph

On a graph of net force versus time, impulse is the signed area between the force curve and the time axis. A force above the axis contributes positive impulse for a chosen positive direction; a force below the axis contributes negative impulse. For a rectangular region, the area is force multiplied by time. For triangles, trapezoids, or more complicated curves, calculate or estimate the total signed area.

This graph rule comes directly from the integral definition of impulse. It is different from work: the area under a force–position graph relates to work, while the area under a force–time graph gives impulse.

The Impulse-Momentum Theorem

Integrating ${\vec{F}}_{n e t} = d \vec{p} / d t$ over time gives

${\vec{J}}_{n e t} = Δ \vec{p} = {\vec{p}}_{f} - {\vec{p}}_{i}$ .

This is the impulse–momentum theorem. It states that the net impulse on an object equals its change in momentum.

Suppose a 0.145 kg ball initially travels in the positive direction at 20 m/s and then leaves a collision at 30 m/s in the opposite direction. With opposite motion represented as −30 m/s,

$Δ p = 0.145 (- 30) - 0.145 (20) = - 7.25 k g m s^{- 1}$ .

The net impulse is therefore −7.25 N s. If the interaction lasts 0.0050 s, the average net force is

$F_{a v g} = \frac{- 7.25}{0.0050} = - 1450 N$ .

The negative sign shows that the impulse and average force point opposite the initially chosen positive direction.

Why Increasing Stopping Time Can Reduce Force

If an object's initial and final momentum are fixed, then its change in momentum is fixed. The required net impulse is therefore fixed as well. Since $J = F_{a v g} Δ t$ , increasing the stopping time can reduce the magnitude of the average force.

This is a key principle behind many safety designs. A seat belt, airbag, helmet padding, gym mat, or crumple zone can lengthen the time over which a person's momentum changes. Real situations are more complex than the simple average-force model, but the impulse idea explains why a longer stopping interval can reduce peak and average forces for the same overall change in momentum.

Datei:Crash-test-with-airbag-and-safty-belt.jpg

Use the crash-test image as evidence for discussion, not as proof by itself. A complete safety analysis must also consider restraint geometry, deformation, force distribution, and human tolerance.

Systems and Conservation of Momentum

System Momentum and External Impulse

For a chosen system,

$Δ {\vec{P}}_{s y s t e m} = {\vec{J}}_{e x t e r n a l}$ .

Internal forces transfer momentum between parts of the system, but equal-and-opposite interaction forces produce canceling internal impulses when the entire interacting system is considered. If the net external impulse is zero or negligible during the interval of interest, then

${\vec{P}}_{i} = {\vec{P}}_{f}$ .

This is conservation of momentum.

The word system is essential. Momentum may change for one object while total momentum of the larger system remains constant. Before solving a problem, decide what belongs inside the system boundary and identify important external forces.

The conservation video shows how Newton's third law and the momentum form of Newton's second law lead naturally to conservation of total momentum for an isolated system.

When Is Conservation a Good Approximation?

A collision may last only a few milliseconds. During that short interval, the impulse from contact forces between the colliding objects can be much larger than the external impulse from gravity or friction. In such cases, total momentum of the colliding system is often approximately conserved during the collision even though external forces are not literally absent.

You should never write "momentum is always conserved" without specifying the system and external impulse. A better statement is: total momentum of an isolated system is conserved, and in many short collisions the external impulse is small enough to neglect.

Collisions

Elastic, Inelastic, and Perfectly Inelastic Collisions

In every collision of an isolated system, total momentum is conserved. What differs is what happens to kinetic energy.

In an elastic collision, total kinetic energy and total momentum are conserved. In an inelastic collision, total momentum is conserved but total kinetic energy decreases because energy is transformed into internal energy, deformation, sound, thermal energy, rotation, or other forms. In a perfectly inelastic collision, the objects stick together and share a common final velocity.

The comparison image is a useful visual overview. Always check the defining equations rather than relying only on visual appearance.

In this animated idealized elastic collision, both total momentum and total kinetic energy are conserved.

In this animated perfectly inelastic example, the carts move together after impact. Total system momentum is conserved, but kinetic energy is not.

This Crash Course video compares collision types and applies momentum ideas in context.

A Perfectly Inelastic Example

An 800 kg cart moves at +4.0 m/s and collides with a 1200 kg cart moving at −1.0 m/s. They stick together. If external impulse is negligible during the collision,

$m_{1} v_{1 i} + m_{2} v_{2 i} = (m_{1} + m_{2}) v_{f}$ .

The initial total momentum is

$(800) (4.0) + (1200) (- 1.0) = 2000 k g m s^{- 1}$ .

The combined mass is 2000 kg, so the final velocity is +1.0 m/s.

The initial kinetic energy is 7000 J, while the final translational kinetic energy is 1000 J. The 6000 J difference is not destroyed; it has been transformed into other forms of energy. This example illustrates why momentum conservation and mechanical-energy conservation are different statements.

Elastic Collisions and Equal Masses

For a perfectly elastic head-on collision between two equal masses, if one object is initially at rest, the moving object can transfer its velocity to the stationary object in the ideal model. This familiar result follows from satisfying both momentum conservation and kinetic-energy conservation.

A Newton's cradle provides a striking demonstration of momentum and energy transfer, though the detailed behavior involves elastic deformation and stress waves through the spheres.

Do not treat the cradle as a chain of perfectly instantaneous two-ball collisions. The real transfer occurs through finite-time deformation and wave propagation.

Two-Dimensional Momentum

Momentum conservation is a vector equation. For an isolated two-dimensional system,

$\sum p_{x, i} = \sum p_{x, f}$

and

$\sum p_{y, i} = \sum p_{y, f}$ .

Choose coordinate axes, resolve each velocity into components, multiply each velocity component by mass, and apply momentum conservation independently in each direction. After solving for final momentum components, you can reconstruct magnitude and direction using vector methods.

This Wikimedia diagram shows a graphical vector solution to a two-dimensional momentum-conservation example. Notice that the vector sum of the momenta is preserved, not the momentum of each object separately.

For two-dimensional work, a sketch is often as important as the algebra. Mark the positive x- and y-directions and label angles relative to a stated axis before inserting trigonometric components.

Problem-Solving Strategy

Choosing the Right Model

Start by deciding whether the question concerns a single object's momentum change or the total momentum of a system. If a force–time relation is given, impulse is often the most direct route. If a short collision occurs within a system whose external impulse is negligible, conservation of momentum is usually central. If the collision is explicitly elastic, you may also use conservation of kinetic energy.

Keep vector directions visible throughout the calculation. In one dimension, assign positive and negative signs before substituting numbers. In two dimensions, work with components. Check units at every major step: momentum uses kg m/s, impulse uses N s, and both units are equivalent.

After obtaining an answer, test whether it is physically plausible. Ask whether its direction matches the expected impulse, whether a final speed is reasonable, and whether any use of kinetic-energy conservation is justified by the collision type.

Common Misconceptions

Momentum is not force. Momentum describes motion and depends on mass and velocity; force describes interaction and changes momentum over time.

Impulse is not simply force. Impulse includes both force and the duration of its action.

An object can have zero net force and nonzero momentum. An object moving at constant velocity has constant momentum even when net force is zero.

Momentum conservation does not require kinetic-energy conservation. Inelastic collisions conserve total momentum for an isolated system while transforming some translational kinetic energy into other forms.

Bouncing usually creates a larger momentum change than merely stopping. If an object reverses direction, the vector change in momentum can have a larger magnitude than if it stops from the same initial velocity.

External force and external impulse are not identical ideas. A nonzero external force acting for a very short time may produce a small external impulse; what matters for momentum change over an interval is the accumulated external impulse.

Laboratory and Data Analysis

A low-friction cart track, dynamics carts, motion sensors, force sensors, video analysis, or a computer simulation can be used to investigate momentum. Before a collision, measure or estimate masses and velocities. After the collision, do the same and compare total momentum within experimental uncertainty.

For an impulse experiment, record force as a function of time during a safe interaction, such as a cart contacting a padded force sensor. Estimate the area under the force–time curve and compare it with the independently measured change in momentum. Differences can arise from sensor calibration, sampling rate, friction, unmodeled external forces, uncertainty in velocity, or an incomplete choice of system.

Never create dangerous high-speed collisions for an experiment. School laboratory carts, soft balls, simulations, or recorded motion can provide the needed evidence safely.

Worked examples are useful for checking whether your sign conventions and system boundaries are consistent.

Summary

Linear momentum is mass times velocity and is a vector. Net force is the rate of change of momentum. Impulse is the time integral of net force and equals the change in momentum. The signed area under a net-force-versus-time graph represents impulse. For a system, change in total momentum equals external impulse. When net external impulse is negligible, total momentum is conserved. Elastic collisions conserve both total momentum and total kinetic energy, while inelastic collisions conserve momentum but not translational kinetic energy. These principles explain collision calculations, recoil, sports impacts, vehicle safety, and many other interactions.

Interactive Tasks

Quiz: Test Your Knowledge

Which expression defines the linear momentum of an object with constant mass? (Mass times velocity) (!Mass times acceleration) (!Force times distance) (!Kinetic energy divided by time)

Which statement correctly describes momentum? (It is a vector in the direction of velocity) (!It is always a positive scalar) (!It has the same unit as energy) (!It depends only on speed)

What does net impulse equal? (Change in momentum) (!Change in kinetic energy) (!Mass divided by velocity) (!Force divided by time)

What does the signed area under a net force versus time graph represent? (Impulse) (!Power) (!Displacement) (!Kinetic energy)

When is total system momentum conserved? (When net external impulse is zero) (!Whenever kinetic energy decreases) (!Whenever objects have equal masses) (!Only when the objects stick together)

What is true in a perfectly inelastic collision of an isolated system? (The objects stick together and total momentum is conserved) (!The objects separate and kinetic energy is conserved) (!Each object keeps its original momentum) (!Total momentum becomes zero)

What is conserved in an elastic collision of an isolated system? (Total momentum and total kinetic energy) (!Only total kinetic energy) (!Only the momentum of each object) (!Neither momentum nor kinetic energy)

Why can an airbag reduce average impact force for the same momentum change? (It increases the stopping time) (!It increases the initial momentum) (!It makes impulse equal to zero) (!It removes the need for a seat belt)

A moving object reverses direction after a collision. What must you do when calculating its momentum change? (Include the change of velocity direction) (!Use speed only and ignore signs) (!Set the final momentum to zero) (!Assume kinetic energy is conserved)

What is the SI unit of impulse? (Newton second) (!Joule second) (!Newton per metre) (!Kilogram per second)

Memory Game

Momentum	Product of mass and velocity
Impulse	Change in momentum caused by net force over time
Isolated system	System with negligible net external impulse
Elastic collision	Interaction conserving momentum and kinetic energy
Perfectly inelastic collision	Interaction after which objects move together
Force-time area	Graphical measure of impulse

Drag and Drop

Match the correct terms.	Topic
Mass times velocity	Momentum definition
Area under a net force-time graph	Impulse from a graph
Equal before and after	Total momentum in an isolated system
Objects move together	Perfectly inelastic collision
Conserved kinetic energy	Elastic collision

...

Crossword Puzzle

Momentum	What vector quantity equals mass times velocity?
Impulse	What vector quantity equals change in momentum?
Vector	What type of quantity has both magnitude and direction?
Elastic	What kind of collision conserves total kinetic energy?
Inelastic	What kind of collision does not conserve total kinetic energy?
Collision	What interaction occurs when objects exert strong forces on each other for a short time?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Momentum photo story: Create a six-image photo story showing everyday situations with different masses and velocities, and add one sentence to each image explaining the expected momentum direction.
Impulse explanation: Write a 250-word explanation for a younger student showing why catching a ball by moving your hands backward can reduce average force.
Force-time sketch: Draw two force–time graphs with the same impulse but different peak forces, label the areas, and explain the safety meaning in a short caption.
Physics interview: Interview a coach, cyclist, mechanic, or science teacher about one situation in which stopping time matters, then compare the interview answer with the impulse–momentum theorem.

Standard

Cart collision experiment: Use safe low-speed carts or a collision simulation to measure masses and velocities before and after a collision, calculate total momentum, and discuss uncertainty.
Momentum video analysis: Record or use a safe video of two low-speed objects interacting, extract approximate velocities from successive frames, and produce a short narrated analysis of momentum before and after.
Safety design poster: Create an evidence-based poster comparing seat belts, airbags, helmets, and crumple zones in terms of impulse, stopping time, force distribution, and system boundaries.
Science museum visit: Visit a science museum, school laboratory, or virtual exhibit that demonstrates collisions, document one apparatus, and explain which momentum assumptions make the demonstration work.

Advanced

Two-dimensional collision investigation: Design a puck or simulation investigation with motion in two dimensions, resolve momentum into components, and present a vector diagram with uncertainty estimates.
Impulse sensor project: Use a force sensor or suitable recorded data to integrate a force–time curve numerically, compare the impulse with measured change in momentum, and explain discrepancies.
Collision model critique: Produce a written or video critique comparing elastic, inelastic, and perfectly inelastic models, including at least one real event that does not fit any ideal model perfectly.
Momentum research presentation: Create a research presentation on a real engineering or sports application of impulse and momentum, use reliable sources, quantify at least one example, and defend the assumptions in your model.

Learning Assessment

System boundary analysis: Given a collision scenario, choose a useful system boundary, identify external impulses, and justify whether momentum conservation is an appropriate approximation.
Force-time evidence: Analyze a non-rectangular force–time graph, calculate impulse from its signed area, and use the result to predict a momentum change with correct direction.
Collision comparison: Compare two collisions with the same initial total momentum but different final kinetic energies, classify each collision, and explain the energy transformations.
Safety transfer task: Evaluate two proposed helmet or packaging designs using stopping time, impulse, and force concepts, and state what additional data you would need before making a recommendation.
Two-dimensional transfer: Solve a two-dimensional collision by components, then explain how the vector result would change if an external impulse acted only in the vertical direction.
Experimental argument: Use a table of measured masses and velocities from repeated cart collisions to decide whether the data support momentum conservation within uncertainty and defend your conclusion.

Evidence of Learning

Knowledge: You can define momentum and impulse, explain the impulse–momentum theorem, state the conditions for momentum conservation, and distinguish elastic, inelastic, and perfectly inelastic collisions.
Skills: You can choose a system, apply vector signs and components, interpret force–time graphs, solve collision equations, check units, and reason with experimental uncertainty.
Products: Strong evidence can include a lab report, annotated graph, vector diagram, simulation study, safety poster, explanatory video, or research presentation with transparent calculations.
Transfer: You can use momentum and impulse to analyze unfamiliar situations in engineering, transport, sports, packaging, robotics, or laboratory physics and can explain the limits of the idealized model.

OERs on the Topic

The English Wikipedia articles on Momentum and impulse provide useful reference material for definitions, historical context, and related mechanics concepts.

Linked Learning Areas

The most important connections are Newton's laws, vectors, energy, collisions, systems, experimental data, and engineering safety. Momentum methods are especially valuable when interaction forces vary rapidly with time and when a collision is easier to analyze as a whole than instant by instant.

Momentum and Impulse