The discussion revolves around understanding the formula for average speed, particularly in the context of multiple segments of travel with varying speeds. The original poster questions how the total time taken can be expressed as the sum of distances divided by their respective speeds, leading to the formula for average speed.
Participants are actively engaged in clarifying the concepts and calculations related to average speed. Some have provided insights into the derivation of time for each segment, while others are seeking simpler explanations and further clarification on specific equations.
There are indications of confusion regarding the definitions of distance and time in the context of round trips, as well as the need for clarity on the variables used in the formulas. Participants are encouraged to articulate their understanding and assumptions more clearly.
What do you mean by "it"? What question does the problem ask?Indranil said:How is it possible?
Perhap you meant ##V_{av} = (s_1 + s_2 + s_3)/ ( s_1/v_1 + s_2/v_2 + s_3/v_3) ##.so, altogether
Vav = s₁ + s₂ + s₃ / ( s₁/v₁ + s₂/v₂ + s₃/v₃)
It means total time taken 's₁/v₁ + s₂/v₂ + s₃/v₃' How is it possible?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##.
Could you get your point a little bit easier, please?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.
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 said:Could you get your point a little bit easier, please?
That's what I want to learn. Please explainDoc 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?
How about this: Distance = Speed X Time.Indranil said:That's what I want to learn. Please explain
Time = distance / speedDoc Al said:How about this: Distance = Speed X Time.
Solve for time.
Now I got it what you mean.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 what about this equation Vav = 2v₁v₂ / (v₁ + v₂) Could you simplify it please?Doc Al said:How about this: Distance = Speed X Time.
Solve for time.
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 said:Now what about this equation Vav = 2v₁v₂ / (v₁ + v₂) Could you simplify it please?
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?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?
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 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?
Vavg = d / tjbriggs444 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.
First, a round trip does not have distance = d. Now write that "t" in terms of d, v1 and v2Indranil said:Vavg = d / t
like this equeation Vavg = t v₁v₂ / t?jbriggs444 said:First, a round trip does not have distance = d. Now write that "t" in terms of d, v1 and v2
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 said:like this equeation Vavg = t v₁v₂ / t?
I think t1 time requiredjbriggs444 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?
One more try and then I'm out. Please write a formula for t1 that uses only d and v1Indranil said:I think t1 time required
Yes t1 = d / v1jbriggs444 said:One more try and then I'm out. Please write a formula for t1 that uses only d and v1
OK. Now repeat for t2, the time for the return trip.Indranil said:Yes t1 = d / v1
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?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.
Indranil said:It means total time taken 's₁/v₁ + s₂/v₂ + s₃/v₃' How is it possible?
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,
ttotIndranil 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?
Vavg = d / t. as t =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.
d(v1 + v2 / (v1v2.) this comes from the adition of (d / v1 + d / v2)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]
No, it isn't.Indranil said:Vavg = d / t.
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.d(v1 + v2) / (v1v2.) this comes from the adition of (d / v1 + d / v2)
You want Vave = Total distance / total time.Indranil said:Vavg = d / t.
Vavg = d1 + d2 / t1 + t2?Doc Al said:You want Vave = Total distance / total time.
Total distance is not equal to 'd'.