How Is Average Speed Calculated for a Round Trip with Different Speeds?

[realslow]

Homework Statement

If you drive from Physicsland to Chemistryland at a constant speed of 120 km/h and immediately return at a constant speed of 60.0 km/h, what is your average speed?

Homework Equations

The Attempt at a Solution

Somebody helped me out on this question and we used a weighted average.
(120 x 1/3) + (60 x 2/3) = 8.0 x 10^1 km/h
The answer is correct, but I don't understand how this weighted averaging works.
It would be great if you could help me understand!

 

[zhermes]

Naively you might guess that the average is 90 km/h right? The problem with that, is that it assumes you were driving for the same period of time at both speeds. In actuality, you were driving the same distance at both speeds. Therefore, you spent less time traveling at 120 km/h than traveling at 60 km/h. Therefore you need to weight 60 km/h more heavily.

Specifically, you were traveling at 60 km/h for twice as long (twice as long in time) as you were traveling at 120 km/h, thus you have to weight 60 by twice as much as 120. Because the total weight must add up to 1, the weights are 2/3 and 1/3.

Another way to think about it: say the distance between locations is 'd.' Solve for how long it takes (in hours) to travel that distance at the given speeds, and weight the speeds by that time (divided by the total time) to find the average.

Does that make sense?

 

[realslow]

Oh, I see!
Yes, it makes sense now :D
Thank you so much!

 

[ashishsinghal]

average speed = net distance / net time

dont confuse with mean of the speeds

 

[rwisz]

I am happy to help explain the concept of weighted average in this context. In order to determine the average speed of your trip, we need to consider the distance traveled and the time it took to cover that distance. Since you traveled at two different speeds, we need to take into account the amount of time spent traveling at each speed.

In this case, we can think of the first leg of the trip (from Physicsland to Chemistryland) as one "part" of the journey, and the return trip (from Chemistryland to Physicsland) as another "part". The distance for each part is the same (since you are traveling the same route), but the time spent traveling is different because of the different speeds.

To calculate the average speed, we need to take into account both parts of the journey and how much time was spent on each part. This is where the concept of weighted average comes in. We assign a weight to each part based on the proportion of time spent on that part. In this case, we spent 1/3 of the total time on the first part (120 km/h) and 2/3 of the total time on the second part (60 km/h). These weights reflect the amount of influence each part has on the overall average speed.

To calculate the weighted average, we multiply each speed by its corresponding weight and then add the results. In this case, it would be (120 km/h x 1/3) + (60 km/h x 2/3) = 8.0 x 10^1 km/h, which is the same as 80 km/h. This means that your average speed for the entire trip was 80 km/h.

I hope this helps to clarify the concept of weighted average in this context. Let me know if you have any further questions.