MATH 221 Instantaneous Velocity Assignment

If you try to define instantaneous velocity you will again end up trying to divide zero by zero. Here is how it goes: When you are driving in your car the speedometer tells you how fast your are going, i.e. what your velocity is. What is this velocity? What does it mean if the speedometer says 50mph?

We all know what average velocity is. Namely, if it takes you two hours to cover 100 miles, then your average velocity was
distance traveled /time it took = 50 miles per hour.

This is not the number the speedometer provides you it doesnt wait two hours, measure how far you went and compute distance/time. If the speedometer in your car tells you that you are driving 50mph, then that should be your velocity at the moment that you look at your speedometer, i.e. distance traveled over time it took at the moment you look at the speedometer. But during the moment you look at your speedometer no time goes by (because a moment has no length) and you didnt cover any distance, so your velocity at that moment is 0 0 , i.e. undefined. Your velocity at any moment is undefined. But then what is the speedometer telling you?

To put all this into formulas we need to introduce some notation. Let t be the time (in hours) that has passed since we got onto the road, and let s(t) be the distance we have covered since then.

Instead of trying to find the velocity exactly at time t, we fifind a formula for the average velocity during some (short) time interval beginning at time t. Well write t' for the length of the time interval.

At time t we have traveled s(t) miles. A little later, at time t + t' we have traveled s(t +t'). Therefore during the time interval from t to t + t' we have moved s(t + t') -s(t) miles. Our average velocity in that time interval is therefore s(t + t') - s(t) /t' miles per hour.

The shorter you make the time interval, i.e. the smaller you choose t', the closer this number should be to the instantaneous velocity at time t.

So we have the following formula (definition, really) for the velocity at time t

v(t) = lim t'+0 s(t + t') -'s(t) /t' .

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