Name: adding velocities across reference frames
Uploaded: 2023-04-30T10:52:43.000Z
Duration: 1 min 11 s
Channel: JoVE

K12 Standards

north-carolina-k-12-science-standard-course-of-study

North Carolina K-12 Science Standard Course of Study

9-12

Physics

motion-and-stability-forces-and-interactions

Motion and Stability- Forces and Interactions

PS.Phy.1.2

Relative Velocity in Two Dimensions

Consider a woman watching an airplane taking off. As the plane moves farther away, the relative motion in the reference frame of the woman can be determined by velocity vector diagrams.
In another example, a boat and a ship are traveling apart from each other with an angle of 45&#176;. When observed from the moving ship, the boat appears to move away at an angle higher than 45&#176;.
The velocity of the boat relative to water is 35 kilometers per hour and the ship&#39;s velocity is 40 kilometers per hour. What will be the velocity of the boat relative to the ship?
Initially, list the known and unknown quantities. Label the vector triangle sketch. Make sure the sum of the angles of the triangle is equal to 180&#176;.
Using the law of sines and by substituting the magnitude of the sides and their opposite angles of the triangle, the velocity of the boat relative to the ship is calculated.

relative velocity in two dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame. The concept of relative velocity can be used to describe motion in two dimensions. Consider a particle P and two reference frames S and S&#8242;. The position of the origin of S&#8242; as measured in S is <img alt="Static equilibrium; &#931;Fx=0; &#931;Fy=0; static equilibrium diagram with force vectors; physics principle" src="/files/ftp_upload/12640/12640img01.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Image01" />, the position of P as measured in S&#8242; is <img alt="Static equilibrium diagram, &#931;Fx=0 and &#931;Fy=0, illustrating forces on a beam and support reactions." src="/files/ftp_upload/12640/12640img02.png" style="opacity: 0.9; height: 18px;" data-altupdatedat="1761830439000" data-alt="Image02" />, and the position of P as measured in S is <img alt="Static equilibrium, &#931;Fx=0 diagram with force vectors and pivot point for physics analysis." src="/files/ftp_upload/12640/12640img03.png" style="opacity: 0.9; height: 18px;" data-altupdatedat="1761830439000" data-alt="Image03" />, which can be evaluated by utilizing vector addition as 
<img alt="Static equilibrium diagram; &#931;Fx=0, &#931;Fy=0 conditions; free body analysis; force balance method." src="/files/ftp_upload/12640/12640_Equation_1.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
Also, the relative velocities are the time derivatives of the position vectors. Therefore,
<img alt="Force equilibrium diagram with free body forces labeled. Static equilibrium, &#931;F=0, vectors shown." src="/files/ftp_upload/12640/12640_Equation_2.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
The velocity of a particle relative to S is equal to its velocity relative to S&#8242; plus the velocity of S&#8242; relative to S. This can be extended to any number of reference frames.
For particle P with velocities <img alt="Static equilibrium beam diagram; &#931;Fx=0, MA=0 equations; tension, weight balance analysis." src="/files/ftp_upload/12640/12640img04.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Image04" />, <img alt="Chromatography diagram, mobile phase flow, stationary phase retention, separation process visualization." src="/files/ftp_upload/12640/12640img05.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Image05" /> and <img alt="Chemical reaction diagram: benzene + bromine &#8594; bromobenzene + HBr, illustrating electrophilic substitution." src="/files/ftp_upload/12640/12640img06.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Image06" /> in frames A, B, and C, the relation can be stated as
<img alt="Chromatography system schematic, DNA separation process, showing columns and elution flow." src="/files/ftp_upload/12640/12640_Equation_3.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation3" />
Also, the relationship between the accelerations observed in two reference frames can be obtained by differentiating the velocity equation
<img alt="Protein purification diagram, chromatography process, illustrating molecular separation stages." src="/files/ftp_upload/12640/12640_Equation_4.png" style="height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation4" />
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/4-5-relative-motion-in-one-and-two-dimensions">Openstax, University Physics Volume 1, Section 4.5: Relative Motion in One and Two Dimensions.</a>

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to ...

Education

JoVE Core

Unit 1

Motion in Two or Three Dimensions

KEY TERMS AND CONCEPTS

adding velocities across reference frames

Relative velocity describes motion as measured from a chosen reference frame. It can also describe the motion of one reference frame compared with another. In two dimensions, this idea helps track how position and velocity change from frame to frame.

Consider a particle P and two reference frames, S and S′. The position of the origin of S′, measured in S, and the position of P, measured in S′, can be combined with vector addition to find the position of P in S. This relation shows how motion in one frame is built from positions in both frames.

Velocity comes from the time derivative of position, so the same frame relation applies to velocities. The velocity of a particle relative to S equals its velocity relative to S′ plus the velocity of S′ relative to S. This rule can be extended to any number of reference frames.

For a particle P with velocities in frames A, B, and C, the same idea links all three frame measurements. The relationship between accelerations in two frames can also be found by differentiating the velocity equation. These vector relations are the basis for analyzing relative motion in one and two dimensions.

Adding Velocities Across Reference Frames

adding-velocities-across-reference-frames

position and displacement vectors

Describing Motion with Position Changes

Describing motion starts with position. Position tells where an object is at a particular time. It must be measured relative to a frame of reference, which is an arbitrary set of axes used to describe position and motion. Earth is often used as a frame of reference. It helps describe an object’s location in relation to stationary objects on Earth. Many important motions also take place in two dimensions, or in a plane. In those cases, position, velocity, and acceleration can each be described...

average and instantaneous velocity vectors

average-and-instantaneous-velocity-vectors

Position Change and Instant Velocity

Position change and instant velocity are key ideas in kinematics. To describe motion, time must be included. Time tells us where an object is and how fast it is moving. The speed of an object comes from how its position changes with time. Each position is tied to a specific time. When the details of motion at each instant are not important, the rate is given as average velocity. Average velocity is a vector. It equals the total displacement between two points divided by the elapsed time. Real...

acceleration vectors

How Velocity Changes Create Acceleration

Acceleration describes how velocity changes over time. In physics, acceleration is a vector, which means it has both size and direction. It points in the same direction as the change in velocity, written as Δv. The larger the acceleration, the larger the change in velocity over a given time. Because velocity is a vector, it can change in more than one way. An object can speed up, slow down, change direction, or do both at once. A runner traveling 10 km/h due east who stops, turns around, and...

direction of acceleration vectors

Negative Acceleration and Motion Reversal

Acceleration describes a change in velocity. That change can be a change in speed, a change in direction, or both. The acceleration vector points in the direction of the velocity change, not always in the direction the object is moving. When an object slows down, its acceleration points opposite to its motion. This is often called deceleration. The term can be confusing in physics because deceleration is not a vector. It does not give a direction relative to a coordinate system, so the correct...

projectile motion

Projectile Paths Under Gravity

Projectile motion describes the path of an object thrown into the air under Earth’s gravity. The object is called a projectile, and its curved path is called the trajectory. This motion appears in many real situations, such as fireworks, a golf ball in flight, meteors entering Earth’s atmosphere, and bullets being fired. When a falling object has no horizontal movement, it is described as one-dimensional projectile motion. When motion includes both horizontal and vertical parts, it is...

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projectile motion: equations

Solving Projectile Motion with Components

Projectile motion can be solved by breaking the motion into horizontal and vertical components. This approach is used for common examples like a basketball throw, an arrow shot from a bow, and kids jumping into a pool. The motion is then analyzed along the x-axis and y-axis. Start by separating displacement into x and y parts. The horizontal and vertical parts of velocity are also separated. If the projectile has speed v and direction θ relative to the horizontal, the vertical component is...

projectile motion: example

Projectile Motion in Sports and Orbit

Projectile motion helps explain both sports throws and motion near Earth. A javelin thrower uses this idea to send the javelin as far as possible. The thrower also takes a short run-up to increase the initial speed of the javelin. For a projectile on level ground, the range is the horizontal distance it travels. The range is largest at a 45° angle, so javelin throwers try to release the javelin as close to that angle as possible. In this case, the range equation assumes the distance is very...

uniform circular motion

Centripetal Acceleration in Circular Motion

Uniform circular motion describes motion in a circle at constant speed. An object follows a curved path, but its speed does not change. The motion of a point on a propeller spinning at a steady rate and the second, minute, and hour hands of a watch are common examples. Even with constant speed, these objects are still accelerating. That idea can seem surprising because in one-dimensional motion, constant speed means zero acceleration. In two- and three-dimensional motion, a particle can...

non-uniform circular motion

Circular Motion with Changing Speed

Circular motion can happen with changing speed. In this case, the object still follows a circular path, but it may speed up or slow down as it moves. When speed changes during circular motion, the motion is called non-uniform circular motion. This adds tangential acceleration, which points along the path of motion and shows a change in speed. Centripetal acceleration is still present in circular motion. It describes the time rate of change of the direction of the velocity vector and points...

relative velocity in one dimension

Relative Motion in a Train and Earth

Relative motion depends on the reference frame used to measure velocity. A reference frame is the point of view from which motion is described. In many examples, Earth is the reference frame. If someone is said to be sitting in a train moving east at 10 m/s, that speed is measured relative to Earth.

Relative velocity describes motion as measured from a chosen reference frame. It can also describe the motion of one reference frame compared with another. In two dimensions, this idea helps track how position and velocity change from frame to frame. Consider a particle P and two reference frames, S and S′. The position of the origin of S′, measured in S, and the position of P, measured in S′, can be combined with vector addition to find the position of P in S. This relation shows how motion...

key terms and concepts

Learn how relative velocity links position, velocity, and acceleration across reference frames in two dimensions with clear vector addition and frame-to-frame motion.

Adding Velocities Across Reference Frames | 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.

Virginia  Standards of Learning

HIgh School

Physical Science

PS.8

PS.8.a

Florida Department of Education

High School

SC.912.P.12

SC.912.P.12.2

Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

Kinematic Equations - I

When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of ...

Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, ...

Use models (graphs, equations, diagrams) to infer motion in one dimension.

PS.Phy.1.1

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

Average and Instantaneous Velocity Vectors

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

Projectile Motion

An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

Projectile Motion: Equations

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

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 ...

Non-uniform Circular Motion

In&#160;uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all ...

Relative Velocity in One Dimension

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a ...

Use models (graphs, equations, diagrams) to infer motion in two dimensions.

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 ...

Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

Use free body models to qualitatively and quantitatively analyze systems of forces in one dimension and two dimensions.

PS.Phy.2.1

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 ...

Weightlessness

When an object is dropped, it accelerates toward the center of the Earth. If the net external force on the object is its weight, it is said to be in free ...

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 ...

/files/media/science-education/science-education-thumbs/14695.jpg

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 ...

first-law-particles-in-one-dimensional-equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton&#39;s first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a ...

first-law-particles-in-two-dimensional-equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic ...

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 ...

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 ...

/files/media/science-education/science-education-thumbs/14697.jpg

Tension

Tension is a force along the length of a medium, in particular, a force carried by a flexible medium, such as a rope or cable. The word &#34;tension&#34; ...

/files/media/science-education/science-education-thumbs/14698.jpg

Carry out investigations to explain the interactions of forces on an object according to Newton's Laws of Motion.

PS.Phy.2.2

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 ...

Dynamics of Circular Motion

An object undergoing circular motion, like a race car, is accelerating because it is changing the direction of its velocity. This centrally directed ...

Dynamics Of Circular Motion: Applications

Suppose a car moves on flat ground and turns to the left. The centripetal force causing the car to turn in a circular path is due to friction between the ...

Use models to qualitatively and quantitatively analyze basic forces related to movement of an object in a circular path (centripetal force).

PS.Phy.2.3

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 ...

/files/media/science-education/science-education-thumbs/14716.jpg

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, ...

Kepler's First Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. He ...

Kepler's Second Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His ...

Kepler's Third Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In ...

Use models to qualitatively and quantitatively explain the relationship among the force of gravity, the distance between two objects, and the mass of the objects, according to the Law of Universal Gravitation.

PS.Phy.2.4

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 ...

Analyze and interpret data to explain the effect of elastic force on objects (Hooke's Law).

PS.Phy.2.5

Analyze the motion of objects using time, distance, displacement, speed, velocity, and acceleration.

Adding Velocities Across Reference Frames

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