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Instantaneous Velocity - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The instantaneous velocity is the velocity of an object at a certain time. If given its position before, during, and after the required time, the instantaneous velocity can be estimated. While estimates of the instantaneous velocity can be found using positions and times, an exact calculation requires using the derivative function. The instantaneous velocity is not the same thing as the average velocity.

I want to talk about the concept
of instantaneous velocity.
Let me step aside for a second and talk about
the difference between instantaneous
speed and average speed.

Let's say you're on a road trip.
You drive 300 miles and it
takes you five hours.
Your average speed is 300 miles divided
by five hours, or 60 miles per hour.
But as you're driving, you look at the speedometer.
It's not telling you average
speed it's telling you instantaneous speed
and it's going to vary a lot.
It may average out to be 60 but it varies
as high as 70, as low as 0 if you stop.
That's the difference between instantaneous
speed and average speed.
Now how do you calculate those things?
Well, that's what we're going
to talk about now.

Let's go back to our pumpkin example.
A pumpkin is catapulted into the air.
Time T is in seconds.
Height is in feet and here's
a small table of values.
Now, suppose I wanted to find the instantaneous
velocity at T equals 0. Well,
I could get a decent approximation by
coming up with the average velocity
over this interval here.
And it would be 200 minus
118, which is 82.
Divided by 1. So 82 feet per second.
I can also use the average velocity from
two to three and that's 250 minus 250
divided by 1. So 50 feet per second.

Now these approximations are using
a change in T of 1. Both of them.
2 minus 1 is 1. Let's see what happens
as we narrow this delta T value.
So here I have a table that includes
T equals 2. But now I have a point
in the left and a point
in the right.
They're much closer
to 2. 1.9.
This is a delta T value of .1.

Now, what are the average velocities?
On the left I get 67.6 feet per second, and
on the right I get 64.4 feet per second.
These values are getting a
lot closer to each other.
Let me go a step further.
Let me go to 1.99 and
Here the delta T value is .01.
And the average velocity on the left is
66.16 and on the right 65.84 to
to the nearest unit
these both round to 66.
So you might say that the instantaneous
velocity is approximately 66 feet per

And it turns out that this is exactly
how we find instantaneous velocity.
As delta T approaches 0, the length of
the time increment that we're taking
average velocity, as that increment goes
to 0, average velocity approaches
the value of instantaneous velocity
at that particular time.

And that's how we calculate
instantaneous velocity.
It's always a limit of average velocities
as delta T goes to 0.

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