You are just getting out of class.

You were walking home when you remembered

that there was a Galaxy Wars marathon on tonight,

so you'd do what every physics student would do: run.

You're pretty motivated to get home,

so say you start running at six meters per second.

Maybe it's been a while since the last time you ran,

so you have to slow down a little bit

to two meters per second.

When you get a little closer to home, you say:

"No, Captain Antares wouldn't give up

"and I'm not giving up either", and you start running

at eight meters per second and you make it home

just in time for the opening music.

These numbers are values of the instantaneous speed.

The instantaneous speed is the speed of an object

at a particular moment in time.

And if you include the direction with that speed,

you get the instantaneous velocity.

In other words, eight meters per second to the right

was the instantaneously velocity of this person

at that particular moment in time.

Note that this is different from the average velocity.

If your home was 1,000 meters away from school

and it took you a total of 200 seconds to get there,

your average velocity would be five meters per second,

which doesn't necessarily equal the instantaneous velocities

at particular points on your trip.

In other words, let's say you jogged 60 meters

in a time of 15 seconds.

During this time you were speeding up and slowing down

and changing your speed at every moment.

Regardless of the speeding up or slowing down

that took place during this path,

your average velocity's still just gonna be

four meters per second to the right;

or, if you like, positive four meters per second.

Say you wanted to know the instantaneous velocity

at a particular point in time during this trip.

In that case, you'd wanna find a smaller displacement

over a shorter time interval

that's centered at that point where you're trying

to find the instantaneous velocity.

This would give you a better value for

the instantaneous velocity but it still wouldn't be perfect.

In order to better zero-in on the instantaneous velocity,

we could choose an even smaller displacement

over that even shorter time interval.

But we're gonna run into a problem here

because if you wanna find a perfect value

for the instantaneous velocity,

you'd have to take an infinitesimally-small displacement

divided by an infinitesimally-small time interval.

But that's basically zero divided by zero,

and for a long time no one could make any sense of this.

In fact, since defining motion at a particular point in time

seemed impossible, it made some ancient Greeks question

whether motion had any meaning at all.

They wondered weather motion was just an illusion.

Eventually, Sir Isaac Newton developed

a whole new way to do math that lets you

figure out answers to these types of questions.

Today we call the math that Newton invented calculus.

So if you were to ask a physicist:

"What's the formula for the instantaneous velocity?",

he or she would probably give you

a formula that involves calculus.

But, in case some of you haven't taken calculus yet,

I'm gonna show you a few ways to find

the instantaneous velocity that don't require

the use of calculus.

The first way is so simple that it's kind of obvious.

If you're lucky enough to have a case

where the velocity of an object doesn't change,

then the formula for average velocity is just gonna give you

the same number as the instantaneous velocity

at any point in time.

If your velocity is changing,

one way you can find the instantaneous velocity

is by looking at the motion on an x-versus-t graph.

The slope at any particular point

on this position-versus-time graph

is gonna equal the instantaneous velocity

at that point in time because

the slope is gonna give the instantaneous rate

at which x is changing with respect to time.

A third way to find the instantaneous velocity is for

another special case where the acceleration is constant.

If the acceleration is constant,

you can use the Kinematic Formulas

to find the instantaneous velocity, v, at any time, t.

(electronic music)