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Instantaneous velocity

Now the idea of average velocity is something that is fairly straightforward, but the idea of instantaneous velocity is a little trickier. It really requires calculus to fully appreciate, but hopefully you already know what a derivative is, so this shouldn't be too hard.

Suppose the velocity of the car is varying, because for example, you're in a traffic jam. You look at the speedometer and it's varying a lot, all the way from zero to 60 mph. What is the instantaneous velocity? It is, more or less, what you read on the speedometer. I'm assuming you've got a good speedometer that isn't too sluggish and can change its reading quite quickly. Your speedometer is measuring the the average velocity but one measured over quite a short time, to ensure that you're getting an up to date reading of your velocity.

So if you measure the displacement of the car over a time , you can use that to determine the average velocity of the car. What we want is to take the limit as goes to zero. More formally, the instantaneous velocity v is defined as

Most of the time we'll be working with instantaneous velocity, so we'll just drop the instantaneous, and call the above v the velocity.

To justify that such a limit exists is something that you've hopefully had to grapple with already. For physics problems, this limit does indeed exist and gives the derivative:

We can go through how this limit works out in the following example.

Joshua Deutsch
Mon Jan 6 00:05:26 PST 1997