31.9. Problems

Today, we will discuss time mean speed. Time mean speed is the average speed of all vehicles passing a specific point over a given period. Can anyone tell me how we might calculate this?

Do we just add up all the speeds?

Good start! Yes, we sum the spot speeds of vehicles and divide by the number of observations. The formula is: $v_t = \frac{\sum v_i}{n}$. Does anyone remember what $n$ represents?

It's the number of vehicles we observed!

Correct! Now let's apply this concept to a quick example. If the speeds are 50, 40, 60, 54, and 45, what would the time mean speed be?

I think it’s $\frac{249}{5} = 49.8$ m/s!

Excellent! You've got it. This is how we can quantitatively evaluate traffic flow.

Let's now move on to space mean speed. This takes into account the time it takes for vehicles to travel a unit distance. How is it different from time mean speed?

Doesn’t it focus more on the spatial distribution of speeds across vehicles?

Exactly! We calculate space mean speed using the harmonic mean of the spot speeds. The formula is $v_s = \frac{n}{\sum \frac{1}{v_i}}$. Can anyone spot the difference here?

It uses the reciprocal of speeds instead of the direct speeds!

Exactly! Remember, space mean speed is always less than or equal to time mean speed. Alright, can someone provide an example calculation?

If the speeds are 50, 40, 60, 54, and 45, then $v_s = \frac{5}{\frac{1}{50} + \frac{1}{40} + \frac{1}{60} + \frac{1}{54} + \frac{1}{45}}$.

That's correct! You’ve understood the concept well.

Great work on the last sessions! Now let's tackle some applied problems. Suppose a frequency distribution shows speeds in intervals; how would we compute the time mean speed?

We first find the average speed for each interval and multiply it by the frequency.

Correct! Then sum those results for the total to calculate time mean speed. Can anyone summarize how we would do this?

We create a table with average speeds, multiplies by frequencies, and sum them up.

Wonderful! This method not only applies to time but can be adapted for space mean speed as well. Now, let’s try an example together based on the problem set.

This sounds challenging and exciting!