Mdthbs

My book states that- "Average speed of a moving body can never be 0 except when time approaches infinity"
I understand that Average speed = (Total dist./ Total time) and that if time is infinity, average speed will be 0 (Anything divided by infinity gives 0)
But I would like to creatively and intuitively understand what does t=infinity mean.

$t=infty$ is a not a useful statement in this context and should be replaced by the time tends towards infinity, $tto infty$, and then the average speed tends towards zero.

So the average speed can get smaller and smaller and smaller but given that you will always be dividing a number (distance travelled) by a very much larger number (time taken) the average speed can never be exactly equal to zero.

First of all, as others have pointed out, $ t=infty$ should be replaced by 'as time approaches infinity' ($t rightarrow infty$).

Now, the average speed will approach zero if the total distance covered doesn't approach infinity as time approaches infinity. For example, if something is moving at a constant speed then as time approaches infinity the distance covered will also approach infinity and the average speed will remain that constant, not zero.

But for the case when total distance covered is finite even in infinite time then surely, it will be moving at a very, very slow speed. Hence it's average speed is also very small, approaching zero.

EDIT: as people have pointed out in the comments, the average speed may approach zero even if distance approach infinity, if it's approaching it slower than time. E.g. $s=sqrt t$

There are a number of ways to answer this, and the answers would depend on how we interpret specific terms. Although we normally use precise meanings sometimes we are not always consistent.

In the book's statement it is clear that if an object moves with constant velocity (being the same speed and direction) then it's average speed will always be the same irrespective of how much time has passed.

So a more precise way of stating what the book means would be to say:

The average speed of an object that has moved can never be 0, but will approach 0 as the time over which we are taking the average speed approaches inifinty.

How are we measuring the time and the distance to identify the average speed? If we have an object in linear motion on the x axis with the displacement from its starting point given by $sin(t)$ then every $pi$ seconds it will be back where it started and at that instant we might legitimately say that it's average speed is 0.

Normally we talk about velocity rather than speed (velocity has a direction (vector value), speed is just an amount (scalar value)) and as velocity (in a linear direction) can be negative it's clear that an average velocity can be zero.

We also might measure distance from the starting point at a particular instant, rather than considering the total distance travelled. In the example where the position is found from $sin(t)$ the position may be the same as the starting point, but the total distance travelled will never be zero.

So an over-simplified statement often raises more questions than it attempts to answer.

Consider an ideal pendulum or an orbit, which both have symmetrical motions. If direction is taken into account (velocity), then everything cancels/balances out. The pendulum bob or planet have a fixed average position. Neither really "goes anywhere" over time, but both certainly have a non-zero average speed.

Velocity is the derivative of displacement (position), and speed is the derivative of the absolute displacement (distance). Taking the absolute value prevents the "balancing" part, so any object that has ever moved will have a non-zero speed from that time forward.

Look at it another way: If an object has moved, when will its average speed be zero? Mathematically, when time approaches infinity. Interpreting the mathematical result as a physical result, you can say that the answer is "never"

A few basic things before we play with the limit of $ttoinfty$. If an object ever moves then the distance traveled by the object has to be a positive finite number, let's say, $s$. By convention, we take that we are analyzing the motion over a period of time starting $t=0$ up until some time $t$. Now, by definition, the average speed is $frac{s}{t}$. As already pointed out by another answer, in an actual experiment, we always deal with a finite value of $t$. Thus, $frac{s}{t}$ is bound to be a finite positive quantity.

However, in physics, there are several situations in which the time over which we observe a system can be very very large as compared to the typical/intrinsic timescale associated with the situation at hand. This motivates us to think carefully about the limit $ttoinfty$ as it can often simplify the problem greatly and/or can reveal interesting features of the system. In short, while it doesn't really make any physical sense to say that $t=infty$, there exist very many motivations (both theoretical and pragmatic) to study the limit $ttoinfty$. So, let's try to analyze what can happen to the average speed, $frac{s}{t}$ in this limit $ttoinfty$.

Case I: $s$ remains finite as $ttoinfty$

This is an easily imaginable case. The particle can either simply stop moving after a certain instance of time, trivially leading to a finite distance traveled even as $ttoinfty$. Or, a particle's motion can be such that it keeps moving more and more slowly as time increases in such a fashion that the distance covered is finite even as $ttoinfty$. For example, if a particle moves according to the equation $x=tan^{-1}t$ then the total distance traveled as $ttoinfty$ is only $frac{pi}{2}$. Thus, the average speed tends to $0$ as $ttoinfty$. Notice that this particle never stops but its speed, $frac{d}{dt}tan^{-1}t=frac{1}{t^2+1}$ keeps decreasing as $ttoinfty$ as expected.

Case II: $s$ also $toinfty$ as $ttoinfty$ but $frac{s}{t}to 0$

The distance traveled by the particle can also approach $infty$ in the limit $ttoinfty$ but if it approaches $infty$ "slower" than $t$ approaches $infty$ then the ratio $frac{s}{t}$ would approach zero. For example, if the motion of a particle is described by $x=sqrt{t}$ then clearly the distance traveled by the particle also diverges as $ttoinfty$. But, since a square-root growth is slower than linear growth, the ratio $frac{s}{t}=frac{sqrt{t}}{t}=frac{1}{sqrt{t}}to 0$ as $ttoinfty$.

Case III: $s$ also $toinfty$ as $ttoinfty$ but $frac{s}{t}$ remains finite

The distance traveled by the particle can approach $infty$ in the limit $ttoinfty$ but if it approaches $infty$ "at a comparable rate" to with which $t$ approaches $infty$ then the ratio $frac{s}{t}$ would remain finite. For example, if the motion of a particle is described by $x=3t$ then clearly the distance traveled by the particle also diverges as $ttoinfty$. But, both $s$ and $t$ grow linearly in $t$, the ratio $frac{s}{t}=frac{{3t}}{t}=3to 3$ as $ttoinfty$.

Case IV: $s$ also $toinfty$ as $ttoinfty$ and $frac{s}{t}toinfty$

The distance traveled by the particle can approach $infty$ in the limit $ttoinfty$ in such a way that it approaches $infty$ "at a faster rate" than with which $t$ approaches $infty$ then the ratio $frac{s}{t}$ would also diverge. For example, if the motion of a particle is described by $x=t^2$ then clearly the distance traveled by the particle diverges as $ttoinfty$. And, since a quadratic growth is faster than a linear growth, the ratio $frac{s}{t}=frac{{t^2}}{t}=tto infty$ as $ttoinfty$.

So, as it happens in Case I and Case II, the average speed can tend to zero as time approaches infinity. But, for any finite interval of time, the average speed cannot be zero as long as the particle has ever moved. Moreover, notice that even in the limit of time approaching infinity, the average speed can also remain finite as in Case III or it can also diverge as in Case IV. Which of these cases will be realized depends on the nature of the motion (trajectory).