The speeds of a group of ten molecules are \(2.0,3.0,4.0, \ldots\), \(11 \mathrm{~km} / \mathrm{s}\). ( \(a\) ) Find the average speed of the group. (b) Calculate the root-mean-square speed of the group.

Short Answer

Expert verified
(a) The average speed of the group is 6.0 km/s (b) The root-mean-square speed of the group is approximately 7.10 km/s.

Step by step solution

01

Calculate Average Speed

The average speed can be calculated by adding up all the molecule speeds, and then dividing by the count of molecules. Here are the speeds: \(2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0\) km/s, totaling ten molecules. Summing these up equals 60 km/s. Thus, the average speed equals \(\frac{60 \text{ km/s}}{10} = 6.0 \text{ km/s}\).
02

Calculate Square of Each Speed

In order to calculate the root-mean-square speed, we first need to find the square of each molecule's speed. The squared speeds are: \(4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0, 100.0, 121.0\) (km/s)\(^2\).
03

Calculate Root-Mean-Square Speed

The root-mean-square speed is found by taking the square root of the average of these squared speeds. Add up the squared speeds to get 505 (km/s)\(^2\), and divide by the number of molecules (10) to find the average, which equals 50.5 (km/s)\(^2\). Taking the square root of this gives approximately 7.10 km/s.

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