Name: instantaneous velocity from position graphs
Uploaded: 2023-04-30T10:50:05.000Z
Duration: 1 min 10 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physical Science

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Motion and Stability- Forces and Interactions

PS.PSc.4.1

Instantaneous Velocity - II

Instantaneous velocity can be calculated from the position versus time graph. Suppose an object moves from point p1 to point p2. Then its average velocity is given by the slope of the position-time graph.
So, if point p2 approaches point p1, and average x-velocity is calculated over shorter time intervals of delta t, then in the limit as delta t tends to zero, the slope of the tangent to the curve at point p1 represents the instantaneous velocity.
If the tangent to the position-time curve slopes upward or downward to the right, its slope, x velocity, and motion are positive or negative accordingly. 
If position as a function of time is known, the instantaneous velocity at any given time can be calculated.
Taking the time derivative of the position function, the velocity as a function of time is estimated. Substituting the value of time, the instantaneous velocity can be obtained.

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 is the limit of the average velocity as the elapsed time approaches zero, or the derivative of displacement with respect to time. Like average velocity, the instantaneous velocity is a vector with the dimensions of length per unit time. Instantaneous velocity can have both positive and negative values. The instantaneous velocity can be calculated from the position-time graph. The instantaneous velocity at a specific time point is the rate of change of the position function, which is the slope of the position function at that point. The slope of the tangent at any point on the position-time graph equals the velocity at that point. A zero slope indicates that the instantaneous velocity is zero. A positive slope indicates positive instantaneous velocity, while a negative slope indicates a negative instantaneous velocity. The steeper the slope (positive or negative), the greater the particle&#8217;s speed in the positive or negative direction.
Consider the motion of a particle in which the position is given by
<img alt="Newton's second law equation, &#931;F=ma, forces diagram with vectors, mechanics concept." src="/files/ftp_upload/12619/12619_Equation_1.png" style="text-align: center; height: 15px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
What is the instantaneous velocity at t = 0.25 s?
The known quantities are the expression for the position as a function of time. To determine the instantaneous velocity, first, consider the equation
<img alt="Static equilibrium diagram; &#931;Fx=0; &#931;Fy=0; force vectors, object, tension analysis, educational use." src="/files/ftp_upload/12619/12619_Equation_2.png" style="text-align: center; height: 30px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
Using the expression of x(t), the expression for velocity is
<img alt="Chaos theory; Lorenz attractor; dynamic systems; butterfly effect; mathematical model." src="/files/ftp_upload/12619/12619_Equation_3.png" style="text-align: center; height: 10px;" data-altupdatedat="1761830439000" data-alt="Equation3" />
Substituting the value for time, in the above equation, the instantaneous velocity at 0.25 s is 1.5 m/s.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/3-2-instantaneous-velocity-and-speed">Openstax, University Physics Volume 1, Section 3.2: Instantaneous Velocity and Speed.</a>

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 ...

Education

JoVE Core

Physics

Unit 1

Motion Along a Straight Line

KEY TERMS AND CONCEPTS

instantaneous velocity from position graphs

Instantaneous velocity describes how fast an object is moving at a specific moment along its path. It is the limit of average velocity as the elapsed time approaches zero. It can also be written as the derivative of displacement with respect to time. Like average velocity, instantaneous velocity is a vector, so it has both size and direction. Its units are length per unit time.

Instantaneous velocity can be positive or negative. A position-time graph can show it at any point on the motion curve. The instantaneous velocity at one time point is the rate of change of the position function. It is also the slope of the tangent line at that point on the graph.

The sign of the slope tells the direction of motion. A zero slope means the instantaneous velocity is zero. A positive slope means positive instantaneous velocity. A negative slope means negative instantaneous velocity. The steeper the slope, the greater the particle’s speed in the positive or negative direction.

To find an instant velocity from a position function, use the expression for position as a function of time. First, write the equation for x(t). Then find the velocity expression from it. For a particle with a given position function, substituting t = 0.25 s gives an instantaneous velocity of 1.5 m/s. This approach matches the definition used for position-time graphs and changing motion.

Instantaneous Velocity from Position Graphs

instantaneous-velocity-from-position-graphs

position and displacement

Reference Frames and Motion in a Line

Position in straight-line motion depends on a frame of reference. A frame of reference is an arbitrary set of axes used to describe an object’s position and motion. It gives meaning to where something is located at a particular time. Earth is often used as the reference frame. In that case, we describe an object’s position relative to stationary objects on Earth. A rocket launch can be described this way. A cyclist’s position can also be described relative to the buildings they pass. <br...

average velocity

Average Velocity and Average Speed

Average velocity and average speed help describe motion in kinematics. To compare motion over time, we use the time variable. Time lets us track an object’s position and how fast that position changes. For motion between two points, each position x_i is paired with a time t_i. When the details of motion at each instant are not important, we use average velocity. Average velocity is a vector quantity, so it includes direction. It is found by dividing total displacement, Δx, by the time...

instantaneous velocity - i

Instantaneous Speed and Direction

Instantaneous velocity describes how fast an object is moving at a specific instant. It gives the motion of a particle at one point in time or at one point along its path. Average velocity over a time interval cannot show this level of detail. Instantaneous velocity, often written as v x , is the limit of average velocity as the elapsed time approaches zero. It can also be found as the derivative of x with respect to t . Like average velocity, it is a vector with dimensions of length per unit...

Instantaneous velocity describes how fast an object is moving at a specific moment along its path. It is the limit of average velocity as the elapsed time approaches zero. It can also be written as the derivative of displacement with respect to time. Like average velocity, instantaneous velocity is a vector, so it has both size and direction. Its units are length per unit time. Instantaneous velocity can be positive or negative. A position-time graph can show it at any point on the motion...

average acceleration

Average Acceleration and Velocity Change

Average acceleration describes how velocity changes over time. In physics, acceleration is not just about speeding up. It also includes slowing down and changing direction. Since velocity is a vector, it has both magnitude and direction. A change in velocity can come from a change in speed, a change in direction, or both. For example, a runner may travel at 10 km/h due east, stop, reverse direction, and then continue at 10 km/h due west. Even though the speed is the same, the velocity has...

instantaneous acceleration

Negative Acceleration and Instant Speed Change

Acceleration shows how velocity changes, and it does not always point in the same direction as motion. When an object slows down, its acceleration points opposite to its motion. This is often called deceleration, but that term can cause confusion because deceleration is not a vector and does not point in a specific direction in a coordinate system. A subway train is a simple example. When the train slows down, it is still accelerating, but the acceleration points opposite to the direction it...

kinematic equations - i

Velocity from Constant Acceleration

Velocity from constant acceleration shows how an object’s speed and direction change when the acceleration stays the same. The first kinematic equation connects initial velocity, acceleration, and time for motion with constant acceleration. It helps describe how velocity changes at a steady rate throughout the motion. The equation matches common sense in several cases. If acceleration is zero, the final velocity is the same as the initial velocity, so the object keeps a constant velocity. If...

kinematic equations - ii

Using the Second Kinematic Equation

The second kinematic equation relates an object’s final position to its initial position, its starting velocity, and the extra distance gained as velocity changes. It is used only when acceleration stays constant during the motion. That makes it useful for straight-line motion problems with a steady rate of acceleration. A car merging onto a 200 m freeway ramp is a good example. In the example, the car starts at 10 m/s and accelerates at 2 m/s². The known values are the distance, the initial...

kinematic equations - iii

Constant-Acceleration Kinematics Formulas

Constant-acceleration kinematics formulas connect an object’s motion when acceleration stays the same. They help describe how velocity and position change over time. These equations are useful for solving motion problems and for understanding the relationships among distance, time, velocity, and acceleration. The third kinematic equation does not use time as a variable. Instead, it gives final velocity in terms of acceleration and the distance over which that acceleration acts. The fourth...

kinematic equations: problem solving

Solving Constant Acceleration Motion Problems

Solving one-dimensional motion problems with constant acceleration starts with identifying the known quantities and choosing the right kinematic equation. A kinematic equation is a motion formula that relates displacement, velocity, acceleration, and time. Depending on what is known and unknown, you may need one equation or two equations to find the answer. In many problems, the number of equations needed matches the number of unknowns. Two-body pursuit problems are different. They always...

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Gravity-Driven Free Fall Motion

Gravity-driven free fall motion is the motion of an object when gravity is the only force acting on it and air resistance is absent. In this case, the object is in free-fall. The acceleration due to gravity stays constant, no matter how much the object weighs. Free-fall is not limited to objects moving straight down. It also includes motion that is affected only by gravity. Because the acceleration is constant, the motion can be described with kinematic equations. Objects dropped from the...

velocity and position by graphical method

velocity-and-position-by-graphical-method

Acceleration Graphs for Motion Prediction

Acceleration graphs can be used to find velocity and position over time. The area under an acceleration-time graph gives the change in velocity. The area under a velocity-time graph gives the change in position, or displacement. A real example is an airplane using an inertial navigation system. Before takeoff, the pilot enters the airplane’s initial position and velocity. The system then uses acceleration data to calculate the airplane’s velocity and position during flight. When acceleration...

velocity and position by integral method

Finding Motion from Acceleration Data

Velocity and position can be found from acceleration using integral calculus. When acceleration is known as a function of time, the velocity function comes from integrating acceleration. The position function then comes from integrating velocity. For constant acceleration, these integral steps match the first and second kinematic equations for velocity and position. The example in this lesson uses a motorboat that is moving at 5.0 m/s and then begins to slow down as it approaches the dock. Its...

key terms and concepts

Learn how instantaneous velocity comes from a position-time graph, tangent slope, and the derivative of position, with a sample value at t = 0.25 s.

Instantaneous Velocity from Position Graphs | JoVE K12

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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.

PS.Phy.1.1

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Ohio’s Learning Standards and Model Curriculum

Forces And Motion

Motion

Motion Graphs

Virginia  Standards of Learning

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PS.8

PS.8.a

PH.2

PH.2.a

Alabama Course of Study Science

Motion and Stability: Forces and Interactions

9-12.P.1.1

9-12.P.1.1.a

Florida Department of Education

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SC.912.P.12

SC.912.P.12.2

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.A

PHYS.5.C

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Georgia Standards of Excellence

SPS8.

Describing Motion

SP1.

One-Dimensional Motion

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Motion Of A Charged Particle In A Magnetic Field

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PS.PSc.6.4

Analyze motion in terms of speed, velocity, acceleration, and momentum.

Instantaneous Velocity from Position Graphs

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