Instantaneous Velocity and Speed

When you drive to Chicago with an average velocity of 50 mph you probably don't drive at this velocity the whole way. Sometimes you might pass a truck and drive at 70 mph and when you get stuck in the traffic jams you might only drive at 20 mph. Now when the police use their radar gun and clock you at 70 mph, you might legitimately protest to the officer that your average velocity for the whole trip was only 50 mph and therefore you don't deserve a speeding ticket. However, as we all know police officers don't care about average velocity or average speed. They only care about your speed at the instant that you pass them. Thus let's introduce the concept of instantaneous velocity and instantaneous speed. What is an instant ? It is nothing more than an extremely short time interval. The way to describe this mathematically is to say that an instant is when the time interval Δt approaches zero, or the limit of Δt as Δt → 0 (approaches zero). We denote such a tiny time interval as dt instead of Δt. The corresponding distance that we travel over that tiny time interval will also be tiny and we denote that as dx instead of Δx. Thus instantaneous velocity or just velocity is defined as:

  (1.1)

Now such a fraction of one tiny dx divided by a tiny dt has a special name. It is called the derivative of x with respect to t. The instantaneous speed or just speed is defined as simply the magnitude of the instantaneous veloctiy or magnitude of velocity.