Revision as of 16:11, 9 April 2017

by Matt Schoonover

edited by Hyunsu Jo

claimed by Justin Duan (Fall 2016)

claimed by Anthony D'Achille (Spring 2017)

Speed and Velocity

Main Idea

Velocity is a vector quantity, meaning that it has both a magnitude and direction. Speed is the magnitude component of a velocity, making it a scalar value. It is important to note that because speed is an inherent component of velocity, you may always calculate the speed of a given velocity; in contrast, given only speed, it is impossible to extrapolate a velocity because 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{ 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.

The concepts of speed and velocity come up often in different areas of study. In CS there are data structures called graphs that are a bunch of vertices connected by edges, these edges can be similar to speed and velocity in that they can have weights and may or may not have directions.

History

Galileo Galilei, an Italian physicist was the person first credited for measuring speed by taking the distance covered and dividing it by the time. He was famous for his falling bodies experiement and discovered that balls of different masses fell at the same speed. He also attempted to measure the speed of light by having two people hold lanterns and see how long it takes for one person to see the other person's light.

References

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

References

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

@@ Line 24: / Line 24: @@
 Speed = <math>\sqrt{V_x^2 + V_y^2 + V_z^2}</math>
-Where V is the velocity vector.
+Where <math>V</math> is the velocity vector.
 Speed can also be considered as <math>\frac{d}{t}</math> in a one dimensional sense where you are only concerned about the object's speed in one dimension.

Speed and Velocity: Difference between revisions

Revision as of 16:11, 9 April 2017

Contents

Speed and Velocity

A Mathematical Model

A Computational Model

Simple Example

Middling Example

Difficult Example

Connectedness

History

See also

External links

References

See also

Further Reading

External links

References

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