Name: adding and subtracting vectors graphically
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K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Forces, Momentum And Motion

Forces in two dimensions

Vector Algebra: Graphical Method

To add vector quantities of the same kind, place the succeeding vector&#39;s tail on the previous vector&#39;s tip. The vector that connects the first vector&#39;s tail with the last vector&#8217;s tip is called the resultant.
Changing the order of vectors does not alter the resultant.
Vectors can be added using the parallelogram rule, which states that if the two vectors&#39;&#160;initial point coincides to form two sides of a parallelogram, then the diagonal from the same point gives the resultant.
For example, consider a boat crossing a river along the northeast direction. If the river flows from west to east, the boat&#39;s actual velocity is the vector sum of both the velocities.
This is given by the diagonal of the parallelogram made by the vectors. The angle of the diagonal gives its direction.
To subtract a vector B from vector A, first find the negative of vector B, and then add it to vector A.
Multiplying a vector by a scalar quantity gives a vector quantity.

vector algebra: graphical method

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
We use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at arbitrary positions. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of the first vector, we draw a line parallel to the second vector. At the end of the second vector, we draw a line parallel to the first vector. In this way, we obtain a parallelogram. From the origin of the two vectors, we draw a diagonal that is the resultant of the two vectors.
The other diagonal of this parallelogram is the vector difference of the two vectors. It follows from the parallelogram rule that neither the magnitude of the resultant vector, nor the magnitude of the difference vector, can be expressed as a simple sum or difference of magnitudes of the vectors. This is because the length of a diagonal cannot be expressed as a simple sum of side lengths. If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultant vectors.
Drawing the resultant vector of many vectors can be generalized using the following tail-to-head geometric construction. We select any one of the vectors as the first vector, and make a parallel translation of a second vector to a position where the origin (&#34;tail&#34;) of the second vector coincides with the end (&#34;head&#34;) of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement. We draw the resultant vector by connecting the origin (&#34;tail&#34;) of the first vector with the end (&#34;head&#34;) of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction.
Scalar multiplication of a vector gives a vector quantity. Depending on the sign associated with the scalar quantity, the direction of the vector is determined. For example, if we multiply the vector quantity with a positive scalar, the new vector will be parallel to the given vector and vice versa.&#160;
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/2-3-algebra-of-vectors">Openstax, University Physics Volume 1, Section 2.3: Algebra of Vectors.</a>

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...

Education

JoVE Core

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

adding and subtracting vectors graphically

Vectors can be added, subtracted, and multiplied by scalars. The vector sum of two or more vectors is called the resultant vector, or resultant. Geometric rules help find the resultant and its direction in a plane.

For two vectors, the parallelogram rule shows how to build the sum. First, move one vector in parallel so both tails start at the same point. Then draw lines through the heads that are parallel to the opposite vector. The diagonal from the shared origin gives the resultant vector.

The other diagonal of the parallelogram gives the vector difference. The size of the resultant or the difference is not a simple sum or subtraction of the two vector lengths. A diagonal is not the same as adding side lengths, so trigonometry is often needed to find the magnitude and direction.

When three or more vectors must be added, the parallelogram rule can be repeated for pairs of vectors. A more general method is the tail-to-head construction. Place the tail of each new vector at the head of the previous one until all vectors are linked in a chain. Then draw the resultant from the tail of the first vector to the head of the last vector.

Because vector addition is associative and commutative, the final resultant does not depend on the order of the vectors in the chain. Scalar multiplication also changes a vector into another vector. A positive scalar keeps the new vector parallel to the original vector, while the sign of the scalar sets the direction.

Adding and Subtracting Vectors Graphically

adding-and-subtracting-vectors-graphically

introduction to scalars

Scalars in Everyday Physics

Scalar quantities are physical quantities that can be described completely by one number and a unit. A class period can last 50 min, a gas tank can hold 65 L, and the distance between two posts can be 100 m. In each case, the value is fully given with no direction needed. The word scalar is a synonym for number. Time, mass, distance, length, volume, temperature, and energy are all examples of scalar quantities. When two scalar quantities have the same units, they can be added or subtracted...

introduction to vectors

Vectors, Scalars, and Direction

Vectors and scalars describe physical quantities in different ways. Some quantities need both magnitude and direction to be fully specified. In those cases, the quantity is called a vector quantity. Examples include displacement, velocity, position, force, and torque. A quantity with only magnitude and no direction is a scalar quantity. The difference matters in everyday examples. If a rescue team knows only the distance to a distress signal, that is not enough to reach it quickly. They also...

vector components in the cartesian coordinate system

vector-components-in-the-cartesian-coordinate-system

Breaking Vectors into x and y Parts

Vectors in a Cartesian coordinate system are often split into x and y parts. This makes it easier to describe direction and size on a rectangular plane. It also matches the way people give directions in everyday life, such as east, west, north, south, and an angle of travel. In a plane, a point is written as an ordered pair, (x, y). A vector is described in a similar way with its vector coordinates. The x-coordinate of the vector is its x-component, and the y-coordinate is its y-component.

polar and cylindrical coordinates

Using Polar and Cylindrical Coordinates

Polar and cylindrical coordinates are useful ways to describe position when rotation or cylindrical shape is involved. Cartesian coordinates work well for displacement, velocity, and forces, but they can become awkward when an object turns or curves around an axis. In the polar coordinate system, a point in a plane is described by two values. The first is the radial coordinate, or the distance from the origin. The second is the angle made by the radial vector, usually measured from the...

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spherical coordinates

Spherical Coordinates for 3D Positioning

Spherical coordinates describe position in three-dimensional space using radius, polar angle, and azimuthal angle. They are especially useful for systems with spherical symmetry, where the shape matches a sphere. Compared with Cartesian coordinates, spherical coordinates can make calculations simpler. To describe a sphere, Cartesian coordinates need three values. Spherical coordinates can use just one parameter, the radius. That is why they are common in science and engineering problems such...

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Vectors can be added, subtracted, and multiplied by scalars. The vector sum of two or more vectors is called the resultant vector, or resultant. Geometric rules help find the resultant and its direction in a plane. For two vectors, the parallelogram rule shows how to build the sum. First, move one vector in parallel so both tails start at the same point. Then draw lines through the heads that are parallel to the opposite vector. The diagonal from the shared origin gives the resultant vector. <br...

vector algebra: method of components

Vector Components for Adding Vectors

Vector components make it easier to add, subtract, and multiply vectors. The parallelogram rule and graphical methods can become hard to use when the math gets complex. One option is to draw vectors to scale, as in navigation, and estimate their lengths and directions from the graph. Another option is the method of components. In many situations, the size and direction of each vector are already known, and the goal is to find the resultant, or combined vector, of several vectors. A real-world...

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scalar product (dot product)

Dot Product and Work in Physics

The dot product links two vectors to a single number. That number is called a scalar. In physics, this idea helps describe work and energy. A common example is work done by a force on an object. Force is a vector, and displacement is also a vector. The work is defined as the scalar product of the force vector and the displacement vector. To find the dot product, multiply the magnitudes of the two vectors by the cosine of the angle between them. The angle can be measured from either vector.

vector product (cross product)

Right-Hand Rule for Vector Multiplication

Vector multiplication gives a vector product, also called the cross product. Its magnitude equals the product of the two vectors times the sine of the angle between them. The result points perpendicular to both vectors. The direction of this perpendicular vector is found with the right-hand thumb rule. Because a plane has two possible perpendicular directions, the rule shows which one to choose. To use it, rotate the first vector about the perpendicular line until it lines up with the second...

scalar and vector triple products

Triple Products in Vectors and Volume

Scalar and vector triple products extend the rules for multiplying two vectors to three vectors. A scalar product gives a scalar, while a vector product gives a vector. These same ideas help define two useful triple-product forms. The scalar triple product is the dot product of one vector with the cross product of two others. It produces a scalar quantity. When the vectors are rotated in a cyclic order, the result stays the same, so this form is associative. It also shows the projection of one...

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gradient and del operator

Gradient Direction on Scalar Fields

The gradient helps describe how a scalar field changes in space. It gives both the size and the direction of the greatest rate of change. In physics and mathematics, this makes it a key tool for studying functions and fields. A simple mountain example shows why direction matters. A slope does not mean much unless it is measured in a specific direction. That is why the directional derivative is used. It is a vector that gives the slope and the direction of change. The gradient of a scalar...

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divergence and curl

Vector Fields: Sources, Sinks, and Rotation

Vector fields can spread outward, pull inward, or rotate around a point. Divergence describes the net outward flow of flux from a small volume through a closed surface as the volume gets very small. In simple terms, it shows how much a vector field spreads out from a point. When the flux moves outward, the divergence is positive. A point with positive divergence is often called a source. When the field contracts toward a point, the divergence is negative. That point is often called a sink.

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second derivatives and laplace operator

Using Del Operators to Build Laplacians

The del operator can form important second-order expressions from gradients, divergence, and curl. These combinations matter in mathematics and physics. They help describe how scalar and vector functions change at a deeper level. For a scalar function, the curl of its gradient is one such expression. For a vector function, the divergence of a curl is another. The curl of a gradient and the divergence of a curl are always zero. The divergence of the gradient of a scalar function leads to the...

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line, surface, and volume integrals

Work, Circulation, and Flux Integrals

Work, circulation, flux, and volume integrals describe how scalar and vector fields add up over a path, surface, or region. For high school science, these ideas help connect math to motion, flow, and stored mass or energy. A line integral for a vector field adds the dot product of a vector function and a tiny displacement vector along a chosen path. If the path is closed, the result is a closed-line integral. The value around a closed path is called the circulation of the vector field. A...

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divergence and stokes' theorems

Divergence and Stokes Theorems in Vector Fields

Divergence and Stokes theorems help connect vector fields to the shapes they pass through. A vector field gives a value with direction at each point, such as electric flow or wind flow. These theorems extend Green's theorem to higher dimensions and also generalize the fundamental theorem of calculus. The divergence theorem focuses on the dot product of a vector field. It lets a difficult surface integral become a volume integral, and it can also work in reverse. With this theorem, the rate of...

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key terms and concepts

Learn how to add and subtract vectors using the parallelogram rule and tail-to-head construction, plus how scalar multiplication affects direction.

Adding and Subtracting Vectors Graphically | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Tennessee Academic Standards

High School

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

PHYS.PS2.4

Florida Department of Education

Physical Science

SC.912.P.12

SC.912.P.12.1

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.B

Georgia Standards of Excellence

SP1.

Adding Vectors

Force

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

In most situations, forces can be grouped into two categories: contact forces and field forces.&#160; Contact forces occur as a result of direct physical ...

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Mass and Weight

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Frictional Force

When a body is in motion, it encounters resistance because the body interacts with its surroundings. This resistance is known as friction, a common yet ...

Static and Kinetic Frictional Force

One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems, and is always in a direction that ...

Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

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Newton's laws applied to complex problems

Content elaboration: Newton's laws of motion, especially the third law, can be used to solve complex problems that involve systems of many objects that move together as one (e.g., an Atwood machine). The equation $a=F_{net}/m$ that was introduced in physical science can be used to solve more complex problems involving systems of objects and situations involving forces that must themselves be quantified (e.g., gravitational forces, elastic forces, friction forces).

newton-s-laws-applied-to-complex-problems

Newton's laws applied to complex problems

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

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Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

/files/media/science-education/science-education-thumbs/14717.jpg

Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Apparent Weight

True weight is the measure of the gravitational force acting on an object. However, if the object accelerates, its measured weight is different from its ...

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Apparent Weight and the Earth's Rotation

Since all objects on the Earth&#39;s surface move through a circle every 24 hours, there must be a net centripetal force on each object, directed towards ...

the-principle-of-superposition-and-the-gravitational-field

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

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Gravitational force and fields

Content elaboration: Gravitational interactions are very weak compared to other interactions and are difficult to observe unless one of the objects is extremely massive (e.g., the sun, planets, moons). The force law for gravitational interaction states that the strength of the gravitational force is proportional to the product of the two masses and inversely proportional to the square of the distance between the centers of the masses, $F_{g}=(G\cdot m_{1}\cdot m_{2})/r^{2}$. The proportionality constant, G, is called the universal gravitational constant and has a value of $6.674\cdot10^{-11}m^{3}/(kg\cdot s^{2})$. Problem solving may involve calculating the net force for an object between two massive objects (e.g., Earth-moon system, planet-sun system) or calculating the position of such an object given the net force. The strength of an object's (i.e., the source's) gravitational field at a certain location, g, is given by the gravitational force per unit of mass experienced by another object placed at that location, $g=F_{g}/m$. Comparing this equation to Newton's second law can be used to explain why all objects on Earth's surface accelerate at the same rate in the absence of air resistance. While the gravitational force from another object can be used to determine the field strength at a particular location, the field of the object is always there, even if the object is not interacting with anything else. The field direction is toward the center of the source. Given the gravitational field strength at a certain location, the gravitational force between the source of that field and any object at that location can be calculated. Greater gravitational field strengths result in larger gravitational forces on masses placed in the field. Gravitational fields can be represented by field diagrams obtained by plotting field arrows at a series of locations. Field line diagrams are excluded from this course. Distinctions between gravitational and inertial masses are excluded. A scale indicates weight by measuring the normal force between the object and the surface supporting it. The reading on the scale accurately measures the weight if the system is not accelerating. However, if the scale is used in an accelerating system, as in an elevator, the reading on the scale does not equal the actual weight. The scale reading can be referred to as the "apparent weight." This apparent weight in accelerating elevators can be explained and calculated using force diagrams and Newton's laws.

Gravitational force and fields

Simple Harmonic Motion

Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke&#39;s law. If the net force can ...

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characteristics-of-simple-harmonic-motion

Characteristics of Simple Harmonic Motion

The key characteristic of the simple harmonic motion is that the acceleration of the system and, therefore, the net force are proportional to the ...

/files/media/science-education/science-education-thumbs/12763.jpg

oscillations-about-an-equilibrium-position

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable ...

Hooke's Law

Hooke&#39;s law states that stress applied to a material is directly proportional to the strain it experiences within the elastic limit. For ductile ...

Stress

When a force is applied on a body, it undergoes deformation. In order to restore the body to its original shape and/or size, an opposite or restoring ...

Strain and Elastic Modulus

The quantity that describes the deformation of a body under stress is known as strain. Strain is given as a fractional change in either length, volume, or ...

Elasticity

Elasticity is the ability of an object to withstand the effects of distortion and to return to its original size and shape once the forces causing ...

Elastic forces

Content elaboration: Elastic materials stretch or compress in proportion to the load they support. The mathematical model for the force that a linearly elastic object exerts on another object is $F_{elastic}=k\Delta x$ where $\Delta x$ is the displacement of the object from its relaxed position. The direction of the elastic force is always toward the relaxed position of the elastic object. The constant of proportionality, k, is the same for compression and extension and depends on the "stiffness" of the elastic object.

Elastic forces

Friction force (static and kinetic)

Content elaboration: The amount of kinetic friction between two objects depends on the electric forces between the atoms of the two surfaces sliding past each other. It also depends upon the magnitude of the normal force that pushes the two surfaces together. This can be represented mathematically as $F_{k}=\mu_{k}F_{N}$, where $\mu_{k}$ is the coefficient of kinetic friction that depends upon the materials of which the two surfaces are made. Sometimes friction forces can prevent objects from sliding past each other, even when an external force is applied parallel to the two surfaces that are in contact. This is called static friction, which is mathematically represented by $F_{s}\le\mu_{s}F_{N}$ The maximum amount of static friction possible depends on the types of materials that make up the two surfaces and the magnitude of the normal force pushing the objects together, $F_{smax}=\mu_{s}F_{N}$ As long as the external net force is less than or equal to the maximum force of static friction, the objects will not move relative to one another. In this case, the actual static friction force acting on the object will be equal to the net external force acting on the object, but in the opposite direction. If the external net force exceeds the maximum static friction force for the object, the objects will move relative to each other and the friction between them will no longer be static friction, but will be kinetic friction.

Friction force (static and kinetic)

Drag Force and Terminal Speed

An interesting force in everyday life is the force of drag on an object when it is moving in a fluid. Like friction, the drag force always opposes the ...

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Density

Density is an important characteristic of substances, crucial in determining whether an object sinks or floats in a fluid. Its SI unit is kg/m3, and its ...

Viscosity

When water is poured into a glass, it falls freely and quickly, whereas if honey or maple syrup is poured over a pancake, it flows slowly and sticks to ...

Air resistance and drag

Content elaboration: Liquids have more drag than gases. When an object pushes on the particles in a fluid, the fluid particles can push back on the object according to Newton's third law and cause a change in motion of the object. This is how helicopters experience lift and how swimmers propel themselves forward. Forces from fluids are quantified using Newton's second law and force diagrams. Factors that affect air resistance and drag and the determination of terminal velocity may be included.

Air resistance and drag

Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...

Work Done Over an Inclined Plane

The center-of-mass framework helps to easily describe the work done on rigid bodies. Since the internal forces in a rigid body do no work, they can be ...

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principle-linear-impulse-momentum-for-single-particle-problem

Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific ...

Forces in two dimensions
• Adding vector forces
• Motion down inclines
• Centripetal forces and circular motion

Content elaboration: Net forces will be calculated for force vectors with directions between $0^{\circ}$ and 360° or a certain angle from a reference (e.g., 37° above the horizontal). Vector addition can be done with trigonometry or by drawing scaled diagrams. Problems can be solved for objects sliding down inclines. The net force, final velocity, time, displacement and acceleration can be calculated. Inclines will either be frictionless or the force of friction will already be quantified. Calculations of friction forces down inclines from the coefficients of friction and the normal force will not be addressed in this course. An object moves at constant speed in a circular path when there is a constant net force that is always directed at right angles to the direction of motion toward the center of the circle. In this case, the net force causes an acceleration that shows up as a change in direction. If the force is removed, the object will continue in a straight-line path. The nearly circular orbits of planets and satellites result from the force of gravity. Centripetal acceleration is directed toward the center of the circle and can be calculated by the equation $ac=v^{2}/r$, where v is the speed of the object and r is the radius of the circle. This expression for acceleration can be substituted into Newton's second law to calculate the centripetal force. Since the centripetal force is a net force, it can be equated to friction (unbanked curves), gravity, elastic force, etc., to perform more complex calculations.

Conservation of Energy

The terms &#39;conserved quantity&#39;&#160;and &#39;conservation law&#39;&#160;have specific scientific meanings in physics, which differ from the ...

Conservation of Energy: Application

When solving problems using the energy conservation law, the object (system) to be studied should first be identified. Often, in applications of energy ...

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

Force and Momentum

Force and momentum are intimately related. Force acting over time can change momentum, and Newton&#39;s second law of motion can be stated in its most ...

Impulse

According to Newton&#8217;s second law of motion, the rate of change of the momentum of an object is the net external force acting on it. The total change ...

Momentum, impulse and conservation of momentum

Content elaboration: Momentum, p, is a vector quantity that is directly proportional to the mass, m, and the velocity, v, of the object. Momentum is in the same direction the object is moving and can be mathematically represented by the equation $p=mv$ The conservation of linear momentum states that the total (net) momentum before an interaction in a closed system is equal to the total momentum after the interaction. In a closed system, linear momentum is always conserved for elastic, inelastic and totally inelastic collisions. While total energy is conserved for any collision, in an elastic collision, the kinetic energy also is conserved. Given the initial motions of two objects, qualitative predictions about the change in motion of the objects due to a collision can be made. Problems can be solved for the initial or final velocities of objects involved in inelastic and totally inelastic collisions. Momentum may be dealt with in two dimensions conceptually, but at this level calculations should be limited to only one dimension. Coefficients of restitution are beyond the scope of this course. Impulse, $\Delta p$ is the total momentum transfer into or out of a system. Any momentum transfer is the result of interactions with objects outside the system and is directly proportional to both the average net external force acting on the system, $F_{avg}$, and the time interval of the interaction, t. It can mathematically be represented by $\Delta p=p_{f}-p_{i}=F_{avg}\Delta t$ This equation can be used to justify why momentum changes due to the external force of friction can be ignored when the time of interaction is extremely short. Average force, initial or final velocity, mass or time interval can be calculated in multi-step word problems. For objects that experience a given impulse (e.g., a truck coming to a stop), a variety of force/time combinations are possible. The time could be small, which would require a large force (e.g., the truck crashing into a brick wall to a sudden stop). Conversely, the time could be extended which would result in a much smaller force (e.g., the truck applying the brakes for a long period of time).

momentum-impulse-and-conservation-of-momentum

Momentum, impulse and conservation of momentum

Adding and Subtracting Vectors Graphically

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