Why total time taken s₁/v₁ + s₂/v₂ + s₃/v₃ in average speed?

• Indranil
In summary, the conversation discusses the equation for average speed and how it is possible to calculate the total time taken for an object to travel a certain distance at different speeds. The formula for average speed is given as total distance divided by total time, and the conversation goes on to explain how to calculate the total time for each segment of the journey. The conversation also discusses a specialized formula for average speed in a round trip scenario, which involves adding the speeds and multiplying them by the distance traveled in each segment. The conversation ends with the attempt to simplify the formula for average speed in a round trip scenario.
Indranil

Homework Statement

As we know the Average speed = Total distance / Total time but why in the case below
Distance traveled = s₁ + s₂ + s₃ and total time taken = s₁/v₁ + s₂/v₂ + s₃/v₃ How is it possible? so, altogether
Vav = s₁ + s₂ + s₃ / ( s₁/v₁ + s₂/v₂ + s₃/v₃)

The Attempt at a Solution

I only know (Average speed = Total distance / Total time)

The total time is the sum of the times for each segment, which you can calculate (as shown) given the speed and distance in each segment.

Indranil said:
How is it possible?
What do you mean by "it"? What question does the problem ask?

so, altogether
Vav = s₁ + s₂ + s₃ / ( s₁/v₁ + s₂/v₂ + s₃/v₃)
Perhap you meant ##V_{av} = (s_1 + s_2 + s_3)/ ( s_1/v_1 + s_2/v_2 + s_3/v_3) ##.

You didn't explain the variables. My guess is that the problem states that an object travels distance ##s_i## at average velocity ##v_i## for ##i =1,2,3##.

Stephen Tashi said:
What do you mean by "it"? What question does the problem ask?Perhap you meant ##V_{av} = (s_1 + s_2 + s_3)/ ( s_1/v_1 + s_2/v_2 + s_3/v_3) ##.

You didn't explain the variables. My guess is that the problem states that an object travels distance ##s_i## at average velocity ##v_i## for ##i =1,2,3##.
It means total time taken 's₁/v₁ + s₂/v₂ + s₃/v₃' How is it possible?

Doc Al said:
The total time is the sum of the times for each segment, which you can calculate (as shown) given the speed and distance in each segment.

Indranil said:
If you're given a distance and speed for an interval, how would you find the time taken to cover that distance at the given speed?

Doc Al said:
If you're given a distance and speed for an interval, how would you find the time taken to cover that distance at the given speed?
That's what I want to learn. Please explain

Indranil said:
That's what I want to learn. Please explain

Solve for time.

Doc Al said:

Solve for time.
Time = distance / speed

Doc Al said:
If you're given a distance and speed for an interval, how would you find the time taken to cover that distance at the given speed?
Now I got it what you mean.

Doc Al said:

Solve for time.

Indranil said:
That's a very specialized formula. Consider something moving from point A to point B at speed v1 and then back to A at speed v2. Now can you figure it out?

Doc Al said:
That's a very specialized formula. Consider something moving from point A to point B at speed v1 and then back to A at speed v2. Now can you figure it out?
Still, I don't understand what you mean. get your point a little bit easier. how do we get '2v₁v₂' in the place of distance and why do we add '(v₁ + v₂)' in the place of time?

Indranil said:
Still, I don't understand what you mean. get your point a little bit easier. how do we get '2v₁v₂' in the place of distance and why do we add '(v₁ + v₂)' in the place of time?
Try following @Doc Al's suggestion. Let d be the distance from point A to point B. Write down a formula for the average velocity: total distance divided by total time. Show us that formula.

jbriggs444 said:
Try following @Doc Al's suggestion. Let d be the distance from point A to point B. Write down a formula for the average velocity: total distance divided by total time. Show us that formula.
Vavg = d / t

Indranil said:
Vavg = d / t
First, a round trip does not have distance = d. Now write that "t" in terms of d, v1 and v2

jbriggs444 said:
First, a round trip does not have distance = d. Now write that "t" in terms of d, v1 and v2
like this equeation Vavg = t v₁v₂ / t?

Indranil said:
like this equeation Vavg = t v₁v₂ / t?
That equation makes no sense. If points A and B are distance d apart, how much time does it take to get from A to B at speed v1?

jbriggs444 said:
That equation makes no sense. If points A and B are distance d apart, how much time does it take to get from A to B at speed v1?
I think t1 time required

Indranil said:
I think t1 time required
One more try and then I'm out. Please write a formula for t1 that uses only d and v1

jbriggs444 said:
One more try and then I'm out. Please write a formula for t1 that uses only d and v1
Yes t1 = d / v1

Indranil said:
Yes t1 = d / v1
OK. Now repeat for t2, the time for the return trip.
Then write down a formula for the total time taken for the round trip.
Finally, write down a formula for the average velocity over the round trip using only d, v1 and v2.

jbriggs444 said:
OK. Now repeat for t2, the time for the return trip.
Then write down a formula for the total time taken for the round trip.
Finally, write down a formula for the average velocity over the round trip using only d, v1 and v2.
t1 + t2 = d / v1 + d / v2, t = d (v1 + v2) / v1v2, Vavg = d / t = d / d(v1 + v2) / v1v2 = v1v2 / v1 + v2 [ but the equeation is 2v1v2 / v1 + v2] Could you check my calculation please?

Indranil said:
It means total time taken 's₁/v₁ + s₂/v₂ + s₃/v₃' How is it possible?

Total time to go distance (s1 + s2 + s3) = time to go s1 + time to go s2 + time to go s3
time to go distance s1 = s1/v1 etc.

Indranil said:
t1 + t2 = d / v1 + d / v2,
This is correct, but the rest of your work is hard to understand. Now write a general expression for the average speed, making use of this expression for the total time.

Indranil said:
t1 + t2 = d / v1 + d / v2, t = d (v1 + v2) / v1v2, Vavg = d / t = d / d(v1 + v2) / v1v2 = v1v2 / v1 + v2 [ but the equeation is 2v1v2 / v1 + v2] Could you check my calculation please?
ttot
= t1 + t2. Yes.
= d/v1 + d/v2. Yes.
= d(v1 + v2) / (v1v2). Where did this come from?

Vavg = d/t Is incorrect. The round trip distance is not given by d.

Doc Al said:
This is correct, but the rest of your work is hard to understand. Now write a general expression for the average speed, making use of this expression for the total time.
Vavg = d / t. as t =
jbriggs444 said:
ttot
= t1 + t2. Yes.
= d/v1 + d/v2. Yes.
= d(v1 + v2) / (v1v2). Where did this come from?

Vavg = d/t Is incorrect. The round trip distance is not given by d.[/sub][/sub]
d(v1 + v2 / (v1v2.) this comes from the adition of (d / v1 + d / v2)

Indranil said:
Vavg = d / t.
No, it isn't.

d(v1 + v2) / (v1v2.) this comes from the adition of (d / v1 + d / v2)
If you skip past the steps where you multiply the first fraction by v2/v2 and the second fraction by v1/v1. combine the terms and factor out the d then yes, that sum follows.

Now what is the round trip distance?

Indranil said:
Vavg = d / t.
You want Vave = Total distance / total time.

Total distance is not equal to 'd'.

Doc Al said:
You want Vave = Total distance / total time.

Total distance is not equal to 'd'.
Vavg = d1 + d2 / t1 + t2?

Indranil said:
Vavg = d1 + d2 / t1 + t2?
What are d1 and d2? We have two points, A and B with distance d between them.

[Note that parentheses matter]

jbriggs444 said:
What are d1 and d2? We have two points, A and B with distance d between them.

[Note that parentheses matter]
Vavg = s / t ?

Indranil said:
Vavg = s / t ?
What's s? We have two points A and B with distance d between them.

jbriggs444 said:
What's s? We have two points A and B with distance d between them.
could you mention the equation, please?

Indranil said:
could you mention the equation, please?

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