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

AI Thread Summary
The discussion centers on understanding the formula for average speed, specifically how total time is calculated as the sum of distances divided by their respective speeds (s₁/v₁ + s₂/v₂ + s₃/v₃). Participants clarify that average speed is derived from total distance divided by total time, emphasizing the need to define variables clearly. The conversation also touches on the average speed formula for round trips, highlighting the importance of correctly applying distance and time calculations. Misunderstandings about notation and the need for clear problem statements are addressed. Overall, the thread emphasizes the foundational concept of average speed as distance over time.
Indranil
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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)

Homework Equations

The Attempt at a Solution


I only know (Average speed = Total distance / Total time)
 
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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.
Could you get your point a little bit easier, please?
 
Indranil said:
Could you get your point a little bit easier, please?
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
How about this: Distance = Speed X Time.

Solve for time.
 
Doc Al said:
How about this: Distance = Speed X Time.

Solve for time.
Time = distance / speed
 
  • #10
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.
 
  • #11
Doc Al said:
How about this: Distance = Speed X Time.

Solve for time.
Now what about this equation Vav = 2v₁v₂ / (v₁ + v₂) Could you simplify it please?
 
  • #12
Indranil said:
Now what about this equation Vav = 2v₁v₂ / (v₁ + v₂) Could you simplify it please?
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?
 
  • #13
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?
 
  • #14
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.
 
  • #15
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
 
  • #16
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
 
  • #17
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?
 
  • #18
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?
 
  • #19
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
 
  • #20
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
 
  • #21
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
 
  • #22
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.
 
  • #23
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?
 
  • #24
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.
 
  • #25
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.
 
  • #26
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.
 
  • #27
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)
 
  • #28
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?
 
  • #29
Indranil said:
Vavg = d / t.
You want Vave = Total distance / total time.

Total distance is not equal to 'd'.
 
  • #30
Doc Al said:
You want Vave = Total distance / total time.

Total distance is not equal to 'd'.
Vavg = d1 + d2 / t1 + t2?
 
  • #31
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]
 
  • #32
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 ?
 
  • #33
Indranil said:
Vavg = s / t ?
What's s? We have two points A and B with distance d between them.
 
  • #34
jbriggs444 said:
What's s? We have two points A and B with distance d between them.
could you mention the equation, please?
 
  • #35
Indranil said:
could you mention the equation, please?
No. That's your job.
 
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  • #36
jbriggs444 said:
No. That's your job.
Vavg = 2d / t?
 
  • #37
Indranil said:
Vavg = 2d / t?
Could anyone clarify me please Vavg = 2d / t? I am waiting for the replies.
 
  • #38
Indranil said:
t1 + t2 = d / v1 + d / v2, t = d (v1 + v2) / v1v2,
Yes

Vavg = d / t
No, in the problem you are considering the total distance is d + d = 2d because you travel distance d from point A to point B and then travel distance d again when you go from point B back to point A.

but the equeation is 2v1v2 / v1 + v2
You are complicating the discussion by writing fractions incorrectly. You omit paretheses that are needed.
You write "2v1v2/ v1 + v2" when you actually mean "2v1v2/ (v1 + v2)"
 
  • #39
Stephen Tashi said:
YesNo, in the problem you are considering the total distance is d + d = 2d because you travel distance d from point A to point B and then travel distance d again when you go from point B back to point A.You are complicating the discussion by writing fractions incorrectly. You omit paretheses that are needed.
You write "2v1v2/ v1 + v2" when you actually mean "2v1v2/ (v1 + v2)"
I got it now. I will check myself. please find my faults so that I can improve myself in Physics. I am learning a lot from this site. one more equation
If t₁ = t₂ = t₃ =...tn, Vav = (v₁ + v₂ +...+ vn)t / nt
What's the concept behind the equation above? Could you clarify it, please?
 
  • #40
Indranil said:
one more equation
If t₁ = t₂ = t₃ =...tn, Vav = (v₁ + v₂ +...+ vn)t / nt
What's the concept behind the equation above?

The proper format of a physics problem is to have a statement of the problem in words. In that statement, the variables are defined. After the variables are defined in words, equations are written.

You give an equation, but you omit the description of the problem in words and you omit defining the variables.

Many of the homework advisors are capable of seeing an equation and guessing what words describe the problem. However, if you rely on other people to "read your mind", you won't learn to write clearly.

Explain the situation and define the variables that are used.
 
  • #41
Stephen Tashi said:
The proper format of a physics problem is to have a statement of the problem in words. In that statement, the variables are defined. After the variables are defined in words, equations are written.

You give an equation, but you omit the description of the problem in words and you omit defining the variables.

Many of the homework advisors are capable of seeing an equation and guessing what words describe the problem. However, if you rely on other people to "read your mind", you won't learn to write clearly.

Explain the situation and define the variables that are used.
Thank you for your kind advice. I'll try to follow.
If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt
What is the concept behind this formula?
 
  • #42
What does 'tn' and 'Vn' mean in the equation below?
'If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt' Could you explain please?
 
  • #43
You have ##n## time intervals, all equally long. ##t_1## is the length of the first interval, ##t_2## is the length of the second one, and so forth.
 
Last edited:
  • #44
Nugatory said:
You have ##n## time intervals, all equally long. ##t_1## is the length of the first interval, ##t_2## is the length of the second one, and so forth.
I don't understand clearly. Could you explain it in detail, please?
 
  • #45
Indranil said:
Thank you for your kind advice. I'll try to follow.
If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt
What is the concept behind this formula?
If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt
What is the concept behind this formula?
I am waiting for the replies
 
  • #46
Indranil said:
If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt
What is the concept behind this formula?
I am waiting for the replies

You have not yet made a complete description of the problem in words. If the problem is written in English, please quote the entire problem. Don't merely quote fragments from it.

If you are translating the problem from another language into English, try to translate each sentence.
 
  • #47
Indranil said:
If t₁ = t₂ = t₃ =...= tn, Then we have Vavg = (v₁ + v₂ +...+Vn)t / nt
What is the concept behind this formula?
The concept behind this formula is the same as all the others: Vave = Distance/Time. Of all the formulas you've been asking about, this basic definition of average speed is the one you need to have down cold. All the rest can be derived from this with a little algebra.

You seem to be obsessing over several formulas that are only useful in very specific situations. (The formula quoted in this post is an example of that -- it's only true under very specific assumptions.) Unless you specify the problem you are trying to solve -- and thus define what the variables mean in your equations -- we can only guess at what you're trying to do. (Admittedly, we are pretty good guessers. But we shouldn't have to guess.)
 
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