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

## 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)

## Answers and Replies

Doc Al
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.

Stephen Tashi
Science Advisor
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##.

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?

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?

Doc Al
Mentor
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?

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

Doc Al
Mentor
That's what I want to learn. Please explain
How about this: Distance = Speed X Time.

Solve for time.

How about this: Distance = Speed X Time.

Solve for time.
Time = distance / speed

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.

How about this: Distance = Speed X Time.

Solve for time.
Now what about this equation Vav = 2v₁v₂ / (v₁ + v₂) Could you simplify it please?

Doc Al
Mentor
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?

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?

jbriggs444
Science Advisor
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.

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

jbriggs444
Science Advisor
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

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?

jbriggs444
Science Advisor
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?

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

jbriggs444
Science Advisor
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

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

jbriggs444
Science Advisor
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.

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?

Stephen Tashi
Science Advisor
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.

Doc Al
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.