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.

Difficult 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] = 33.17m/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.

References

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

Simple Examples

1. If a person walked a distance of 10 meters in 5 seconds, what is their average velocity?

Solution: Using the average velocity equation will tell you that the answer is 2 m/s.

2. If someone traveled with a velocity of 13 m/s for 3 minutes, how far would they have traveled?

Solution: First convert 3 minutes to seconds(180). Then use the velocity equation to solve for distance. You should get 2,340 m.

3. What is the unit vector for the velocity vector <3,4,0> m/s?

Solution: Use the speed equation to get that the speed is 5 and the unit vector equation to get a final answer of <3/5,4/5,0>

4. What is the velocity of an object if the unit vector is <3,2,1> and the speed is 5 m/s?

Solution: 5 m/s * <3,2,1> = <15,10,5> m/s

References

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

Speed and Velocity

Contents

Speed and Velocity

A Mathematical Model

A Computational Model

Simple Example

Difficult Example

Connectedness

See also

External links

References

Simple Examples

See also

External links

References

Navigation menu