Speed and Velocity

by Matt Schoonover

edited by Hyunsu Jo

claimed by Justin Duan (Fall 2016)

Speed and Velocity

Main Idea

Velocity is a vector quantity, meaning that is has both magnitude and momentum. Speed is the magnitude of a velocity, meaning it's only a scalar without direction. Objects with velocities inherently have a speed and objects with speed don't necessarily have to have a velocity if no direction is specified.

A Mathematical Model

Speed

Speed = [math]\displaystyle{ \sqrt{V_x^2 + V_y^2 + V_z^2} }[/math]

Where [math]\displaystyle{ \boldsymbol{\vec{v}} }[/math] is the velocity vector.

Speed can also be considered as [math]\displaystyle{ \frac{d}{t} }[/math] in a one dimensional sense where you are only concerned about the object's speed in one dimension.

Velocity

Velocity = [math]\displaystyle{ \boldsymbol{\vec{v}} = \frac{\Delta\vec{r}}{\Delta{t}} }[/math]

Where [math]\displaystyle{ {\Delta\vec{r}} }[/math] is the change in the position of the object and t is the time period over which the change in position occurred.

A Computational Model

Difference between speed and velocity could be easily explained by the pictures of two computational models.

In the first computational model, the speed which is a scalar quantity(one dimension) is only in x-direction.

However, in the second computational model, the velocity is in both x and y direction which is in 3 Dimensional <x,y,z> plane which differentiates speed and velocity.

Simple Example

1) What is the average speed of a person who walked a distance of 15 meters in 3 seconds?

This is a problem where we only care about a single dimension, in this case the direction is not specified but is assumed that it's in along the path of the distance.

Using the formula, speed = [math]\displaystyle{ \frac{d}{t} }[/math], where d is 15 meters and t is 3 seconds, the speed is 15/3 = 5 meters per second.

2) Find the velocity of a person who traveled from <0, 0, 10> to <10, 10, 20> in 5 seconds.

Using the formula [math]\displaystyle{ \boldsymbol{\vec{v}} = \frac{\Delta\vec{r}}{\Delta{t}} }[/math], where [math]\displaystyle{ {\Delta\vec{r}} }[/math] is <10, 10, 20> - <0, 0, 10> = <10, 10, 10> and where [math]\displaystyle{ {\Delta{t}} }[/math] is 5,

<10, 10, 10> / 5 = <2, 2, 2>. So the velocity of the person is <2, 2, 2> m/s.

Middling Example

1) At time 0.3 seconds after the tennis ball has been hit by a tennis racket, the tennis ball is located at <4,7,3>m, relative to an origin of the tennis court. At time 0.6 seconds after the tennis ball got hit, it was located at <13, 10, 6>m.

What is the velocity of the tennis ball?

Use the formula: [math]\displaystyle{ \boldsymbol{\vec{v}} = \frac{\Delta\vec{r}}{\Delta{t}} }[/math]

[math]\displaystyle{ {\Delta\vec{r}} }[/math] = <13, 10, 6> - <4, 7, 3> = <9, 3, 3> [math]\displaystyle{ {\Delta{t}} }[/math] = .6 - .3 = .3. <9, 3, 3>/ .3 = <30,10,10> m/s

What is the speed of the tennis ball?

Use the formula: [math]\displaystyle{ \sqrt{V_x^2 + V_y^2 + V_z^2} }[/math]

speed = [math]\displaystyle{ \sqrt{30^2+10^2+10^2} }[/math] = 33.17 m/s

Difficult Example

What is the velocity of an object if the object is traveling in the direction <.566, .793, -.226> at a speed of .04416 km/s?

Knowing the speed is the magnitude of velocity and that velocity is a vector, the original vector can be found by multiplying the magnitude of the vector by the unit vector. Convert km/s to m/s first, .04417 km/s * 1000 m/ 1 km = 44.16 m/s

<.566, .793, -.226> * 44.16 = <24.55, 35.02, -9.98> m/s

Connectedness

Both speed and velocity represents speed of an object. This is why speed and velocity are often interchangeably used in incorrect ways. Speed represents one dimensional speed of an object while velocity represents three dimensional speed of an object with directions of <x,y,z> plane.

See also

• Vectors
• Velocity
• Newton’s First Law of Motion

External links

• Physics lecture resources by Khan academy [1]
• Physics lecture resources by Physics classroom [2]

References

Matter and Interactions, Modern Mechanics, 3rd Edition by Chabay and Sherwood.

See also

• Kinds of Matter
• Detecting Interactions
• Fundamental Interactions
• System & Surroundings
• Newton's First Law of Motion
• Newton's Second Law of Motion
• Newton's Third Law of Motion
• Gravitational Force
• Terminal Velocity and Friction Due to Air
• Simple Harmonic Motion
• Electric Polarization

External links

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• A physics resource written by experts for an expert audience Physics Portal
• A wiki book on modern physics Modern Physics Wiki
• The MIT open courseware for intro physics MITOCW Wiki
• An online concept map of intro physics HyperPhysics
• Interactive physics simulations PhET
• OpenStax algebra based intro physics textbook College Physics
• The Open Source Physics project is a collection of online physics resources OSP
• A resource guide compiled by the AAPT for educators ComPADRE

References

Matter and Interactions, Modern Mechanics, Volume One 4th Edition by Chabay and Sherwood.