Why average speed is important?

Question

We know that in kinematics we have the concepts about "average speed".By definition the average speed is the total of the distance divided by time , but i still don't get it what is the average speed intuitively and why average speed is so important , and in my opinion average speed it doesnt seems accurate , for example if you say the average 12 km/h , it doesnt mean that you constantly Drive with 12 km/h . I am still confuse what average speed is actually mean and why do we need to learn it?

Answer 1

It is a term to learn because it is a term which is used. Of course there are different tools for analysing "speed" and "velocity" - not least differential calculus, as well as statistical tools for modelling variability.

But, for example, the (very useful) Mean Value Theorem in calculus relates to a global average.

And if you are looking for a value to use as a baseline to simplify equations, sometimes analysing around the average rather than around some other "zero" can be helpful.

Answer 2

If your average speed is 12 km/h it doesn't really mean you that you constantly drive at 12 km/h indeed. But it does tell that if you were driving at constant speed, this would be the speed necessary to complete the route.

The average speed is useful because average values of quantities are usually useful in applications. However, I find it interesting because it can help you built up a certain intuition about the definition of instantaneous speed/velocity. Take for instance the definition: $$v(t) := \lim_{\Delta t \to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}.$$ The object: $$\frac{x(t+\Delta t)-x(t)}{\Delta t}$$ is simply the average speed of going from $x(t)$ to $x(t+\Delta t)$ during the time interval $\Delta t$. So, the definition of $v(t)$ is actually telling you that the instantaneous speed is obtained by calculating the average speed when the time interval $\Delta t$ goes to zero.

Answer 3

Mathematically, average speed is not important. (But see below!)

Instead, we care a lot about the mean value theorem, because it lets us estimate long-term quantities in terms of derivatives. The classic application of this is deriving the Lagrange error bound for Taylor series. It also is very common in differential equations theory; I believe it is used to prove (for example) that the $n$^th Sturm-Liouville eigenfunction has $n-1$ zeroes, although the argument escapes me at the moment.

Pedagogically, asking about average speed is a good way to motivate the mean value theorem for students who have not yet seen either of the above applications.

Physically, average speed is extremely important. We can't measure anything instantaneously, so we can't measure instantaneous velocities. Instead, we can measure average speeds; the fact that these converge to a well-defined velocity as we shrink the measurement interval is a hidden assumption of classical mechanics. Average speed thus provides a way to tether our theoretical quantities to observed phenomena.

Answer 4

You said "if you say the average 12 km/h , it doesnt mean that you constantly Drive with 12 km/h ". So if some one ask you:

• "How fast were you traveling in your trip?"

Will you answer like this?

• " I begin at $0$ km/h, accelerated to $30$ km/h in the following $30$ seconds, then to $50$ km/h in the following $6$ minutes, then I stoped in a red light, so that were $0$ km/h for $45$ seconds, ... (continue with this description for the next $5$ hours)".

When you're done with your answer the person who ask you will be really tired and will probably had no idea if your trip was fast or slow.

Or you rather say this?

• "Rather slowly, $60$ km/h in average"