Problem 78 - General Chemistry: Principles and Modern Applications

Question

Determine \(u_{\mathrm{m}}, \bar{u},\) and \(u_{\mathrm{rms}}\) for a group of ten automobiles clocked by radar at speeds of 38,44,45,48,50 \(55,55,57,58,\) and \(60 \mathrm{mi} / \mathrm{h},\) respectively.

Step-by-Step Solution

Answer

The mean speed of the vehicles \(\bar{u}\) is 51.0 mi/h, the mode of the speeds \(u_{\mathrm{m}}\) is 55 mi/h, and the root mean square speed \(u_{\mathrm{rms}}\) is 52.39 mi/h.

Step 1: Calculation of Mean \(\bar{u}\)

The mean \(\bar{u}\) for the speed of cars is defined as the sum of all speed values divided by the number of values. That is: \(\bar{u} = \frac{1}{n} \sum_{i=1}^{n} u_i\). So, add up the speeds: 38 + 44 + 45 + 48 + 50 + 55 + 55 + 57 + 58 + 60 and then divide by 10.

Step 2: Calculation of Mode \(u_{\mathrm{m}}\)

The mode \(u_{\mathrm{m}}\) is the speed value that appears most frequently in the dataset. If you look at the dataset, it shows that the speed value 55 appears twice. Therefore the mode of this set is 55.

Step 3: Calculation of Root Mean Square \(u_{\mathrm{rms}}\)

The root mean square \(u_{\mathrm{rms}}\) for the speed of cars is defined as the square root of the mean of the squares of all the values. That is: \(u_{\mathrm{rms}} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} u_i^2}\). So square each speed: \(38^2, 44^2, 45^2, 48^2, 50^2, 55^2, 55^2, 57^2, 58^2, \) and \(60^2\), then add these results together, divide by 10 and then take the square root of the result.

Key Concepts

Mean Calculation

In statistics, the mean is a measure of central tendency commonly referred to as the "average." It gives us an idea of the central value around which other data points congregate. Imagine it as the balancing point of a seesaw. To calculate the mean, sum up all the individual data points and then divide by the number of points. For the provided speeds, you calculate the mean (\(\bar{u}\)) by adding each speed:

• 38 + 44 + 45 + 48 + 50 + 55 + 55 + 57 + 58 + 60.

Next, divide the total by the number of cars, which is 10. So the mean speed is:\[\bar{u} = \frac{38 + 44 + 45 + 48 + 50 + 55 + 55 + 57 + 58 + 60}{10}\]Calculate the total: 510, and then divide by 10 to find the average speed of 51 mi/h.
This tells us that, on average, the cars are moving at a speed of 51 mi/h.

Mode Determination

The mode is another measure of central tendency. However, unlike the mean, the mode tells us the most frequently occurring value in a dataset. It can be quite useful when data contains outliers or when it’s important to know what is common within a dataset. For the provided group of car speeds, to find the mode, look for the number that appears the most.

• In our dataset, 38, 44, 45, 48, 50, 55, 55, 57, 58, and 60, notice that the number 55 appears twice.
• Other numbers appear only once.

Thus, 55 is the mode. Knowing the mode gives us insight into what speed you might expect to encounter most frequently among the cars clocked.

Root Mean Square Calculation

The root mean square (RMS) is a measure used in statistics to provide the square root of the average of the squares of the values in a dataset. It's particularly useful in physics and engineering because it treats all deviations, positives or negatives, equally. To calculate the RMS speed for the given data, follow these steps:

• First, square each speed: \(38^2, 44^2, 45^2, 48^2, 50^2, 55^2, 55^2, 57^2, 58^2, \) and \(60^2\).
• Add these squared values together.
• Divide the sum by the number of values (10 in this case).
• Finally, take the square root of the result.

This can be mathematically expressed as:\[u_{\mathrm{rms}} = \sqrt{\frac{1}{10} (38^2 + 44^2 + 45^2 + 48^2 + 50^2 + 55^2 + 55^2 + 57^2 + 58^2 + 60^2)}\]The result will yield the RMS speed. This metric helps in understanding the distribution of speeds in a way that the mean might not, as it accounts for variability in a more pronounced manner.