Quant Exotica- Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance

Here is a textbook situation in Time, Speed and Distance: A man goes from point A to point B with velocity v₁ and returns with velocity v₂. What is his average velocity?

Using the formula , the average velocity can be found to be . So far so good.

Ever wondered why we get this result? Why do we get the velocity as the harmonic mean of the two velocities? The answer lies in basics of arithmetic and harmonic progressions.

Let t₁ and t₂ be the time taken while going from A to B and coming back. The situation is shown below:

Now what will a hypothetical average velocity in this situation mean? It will mean that a person takes the same time, t_average, while going from A to B and coming back. The situation is summarized below:

The total time taken will be same as t₁ + t₂

--> 2 × t_average = t₁ + t₂ or t_average = (t₁ + t₂)/2

In another words, t₁, t_average, and t₂ will be in arithmetic progression.

So how is this related to velocity?

Remember that when distance is constant velocity is inversely proportional to time?

I.e. V is proportional to 1/T or V = k/T.

If T₁, T₂, and T₃ were in arithmetic progression, then 1/T₁, 1/T₂, 1/T₃ are in harmonic progression => k/T₁, k/T₂, k/T₃ are in harmonic progression => V₁, V₂ and V₃ are in harmonic progression!

It can also be proved that if V₁, V₂ and V₃ are in arithmetic progression, then T₁, T₂, and T₃ are in harmonic progression.

So here’s the rule:-

Now let’s apply these rules in practical problems:-

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	Re: Quant Exotica- Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance by rmmozhi prathiba - Tuesday, 8 May 2007, 11:20 PM
	A and S walk up an escalator. The escalator moves at a constant speed A takes 9 steps for every 16 of S's steps. A gets to the top of the escalator after having taken 30 steps while S, because of his faster pace, ends up taking 40 steps to reach the top. If the escalator was turned off, how many steps wud they have to take to walk up? how to answer this tg??? help me plz.. Show parent | Reply

	Re: Quant Exotica- Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance by Total Gadha - Wednesday, 9 May 2007, 03:09 AM