Calculating Average Speed: Point 1 to Point 2 and Back, Using Identical Path

In summary, the average speed for a round trip from point 1 to point 2 and back using the identical path is given by the formula 2S1S2/(S1+S2), where S1 is the average speed from point 1 to point 2 and S2 is the average speed from point 2 back to point 1. This is the harmonic average, not the arithmetic average, as the time taken to go both ways is equal.
  • #1
thursdaytbs
53
0
You move from point 1 to point 2, and then from point 2 back to point1, using the identical path. if your average speed from point 1 to point 2 is S1 and the average speed from point 2 back to point 1 is S2. What's the average speed of the entire trip there and back, in terms of S1 and S2.

Wouldn't the average speed going to point 2, and coming back to point 1 sum divded 2, basically (S1+S2)/2 be the average speed for the whole trip?
 
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  • #2
Yes that is correct.
 
  • #3
But the problem should not be this easy. Our teacher had said that it's not what you think it is.

Therefore, I had come up with this, which doesn't come out to make sense for some reason. Maybe someone could point out why?

Average Speed = Distance Traveled / Time it took to travel

S1 = (1->2) / t1 and s2 = (2->1) / t2
*1->2 = 2->1, let's just call them X
S1 = X / t1 and s2 = X / t2

Total average speed = [(1->2) + (2->1)] / (t1 + t2)
= (X+X) / (t1+t2)
= 2X / (t1+t2)

now... t1 = X/S1 and t2 = X/S2
so...

2X / (X/S1 + X/S2) turns out to equal (2)(S1)(S2) / (S1+S2)

Would that be the correct answer? But... If I say S1 = 2, and S2 = 4, then the average speed should be (S1+S2)/2 = 3. But.. using (2)(S1)(S2) / (S1+S2) you get, 8/3... ?

Any help appreciated.
 
  • #4
i think your method is the correct one since you derived it from the definition of average speed.

average speed is not the same as the average in statistics ie arithmetic mean.
 
  • #5
mattmns said:
Yes that is correct.


NO, that's completely wrong! "average speed" is not "arithmetic average of two numbers". It is, in fact, the "harmonic average".

The "average" speed is defined as the speed such that one would cover that distance in exactly the same time as with the varying speed.

Let D be the distance from point 1 to point 2. Then at speed S1, it would take D/S1 (time units). Going back at speed S2, it would take D/S2. The time to go both ways is D/S2+ D/S1, of course.

Let S be the average speed. To go the total distance 2D at speed S would require 2D/S (time units). Those two times must be the same:

2D/S= D/S1+ D/S2.

Fortunately, the "D" cancels out. 2/S= 1/S1+ 1/S2. Multiplying through by S1, S2, and S to get rid of the fractions, 2S1S2= S2S+ S1S= (S2+ S1)S so
S= 2S1S2/(S1+ S2).

Hey, that's what thursdaytbs got!

(Unfortunately, you spoil it by saying "But... If I say S1 = 2, and S2 = 4, then the average speed should be (S1+S2)/2 = 3." Didn't you start by saying you didn't think it was just the arithmetic average of two numbers!)
 

FAQ: Calculating Average Speed: Point 1 to Point 2 and Back, Using Identical Path

How do you calculate average speed for a round trip using the same path?

To calculate average speed for a round trip using the same path, you need to find the total distance traveled and the total time taken. Then, divide the total distance by the total time to get the average speed.

What is the formula for calculating average speed for a round trip?

The formula for calculating average speed for a round trip is: average speed = total distance / total time. Make sure to use the same units for distance and time (e.g. kilometers and hours).

Can you use different units for distance and time when calculating average speed?

No, you cannot use different units for distance and time when calculating average speed. To get an accurate result, both distance and time should be in the same units. If necessary, you can convert the units before calculating the average speed.

Is average speed the same as constant speed?

No, average speed is not the same as constant speed. Average speed is the total distance divided by the total time, while constant speed is the speed maintained throughout the entire journey. Average speed can be calculated even if the speed varies during the journey, but constant speed cannot.

How can you use the concept of average speed in real-life situations?

The concept of average speed can be used in real-life situations to calculate the average speed of a vehicle during a road trip, the average speed of a runner during a race, or the average speed of a chemical reaction in a laboratory. It can also be useful for estimating travel time and fuel consumption.

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