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.

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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

2.01.

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

second.

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.