# Instantaneous speed and velocity

## Video transcript

- [Instructor] Pretend
you are a physics student. 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 whether
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)