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

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)

Mentor
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.

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##.

Indranil
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?

Indranil
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.

Mentor
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?

Indranil
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

Mentor
That's what I want to learn. Please explain

Solve for time.

Indranil

Solve for time.
Time = distance / speed

Indranil
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.

Indranil

Solve for time.

Mentor
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?

Indranil
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?

Homework Helper
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.

Indranil
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

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

Indranil
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?

Homework Helper
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?

Indranil
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

Homework Helper
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

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

Homework Helper
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.

Indranil
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?

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.

Mentor
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.

Homework Helper
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.

Indranil
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 =
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)

Homework Helper
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?

Mentor
Vavg = d / t.
You want Vave = Total distance / total time.

Total distance is not equal to 'd'.

Indranil
You want Vave = Total distance / total time.

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

Homework Helper
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]

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

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

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

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

• 