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.