Why Is the Average Speed Not Simply the Arithmetic Mean of Two Speeds?

[marco101]

I just wanted to know how to find average speed without knowing the distance. I know average speed is defined as total distance / change in time but what would you do
a person travels to a friend's house and when he is halfway there, he figures out that he has been averaging 30 mph. He drives the last half of the trip at 60 mph.
What is Joe’s average speed for the entire journey?

I figured it was just 30 + 60 divided by 2 but apparently i am wrong. anyone can help?

 

[n1person]

It is not quite that linear because you cover more per time going 60 miles per hour than 30 miles per hour.

We will use dimensional analysis just to make sure our operations are making sense.

60 (m/h) * x (h) + y (m/h) * x (h) = c miles

we also know that 30 (m/h) = c (m) / 2x (h)

so 2x (h) = c (m) / 30 (m/h)

So now you can solve for x, the c's will cancel out (cause it doesn't depend on the distance traveled), solve then for y, and you'll get the answer.

 

[tomcue]

Hello there,

Thank you for your question. Finding average speed without knowing the distance can be a bit tricky, but it is definitely possible. In this scenario, we have a person who travels to their friend's house and halfway through the trip, they realize they have been averaging 30 mph. For the second half of the trip, they drive at 60 mph. The question is, what is the person's average speed for the entire journey?

The formula for average speed is indeed total distance divided by change in time. However, in this case, we do not have the total distance. So, we need to use a different approach. We can use the concept of weighted average to find the average speed.

To calculate the weighted average, we need to multiply each speed by the time it was traveled and then divide the sum of these values by the total time traveled. In this scenario, the first half of the trip was traveled at 30 mph for half the time, and the second half was traveled at 60 mph for the other half of the time.

So, the calculation would be: (30 mph x 0.5 hours) + (60 mph x 0.5 hours) / (0.5 hours + 0.5 hours) = (15 mph + 30 mph) / 1 hour = 45 mph / 1 hour = 45 mph

Therefore, Joe's average speed for the entire journey is 45 mph.

I hope this helps clarify how to find average speed without knowing the distance. If you have any further questions, please feel free to ask. I am always happy to help with any scientific inquiries.

Best regards,