Question

Six molecules speeds are 2 units, 5 units, 3 units, 6 units, 3 units and 5 units respectively. The rms speed is:
A. 4 units
B. 1.7 units
C. 4.2 units
D. 5 units

Answer 1

Hint: The root mean squared or rms velocity is the square root of average squared velocities. We can use this definition and find the rms velocity by using a formula based on this definition.
Formula used:
If we have n different velocities, then the rms velocity for the distribution is given by:
$v_{rms} = \sqrt{ \dfrac{v_1^2 + v_2^2 +... + v_n^2}{n}}$

Complete answer:
We are given six different velocities with magnitudes 2 units, 5 units, 3 units, 6 units, 3 units and 5 units. We first find the squares of these velocities and add them respectively as:
4 + 25 + 9 +36 + 9 + 25 = 108.
This will give us the numerator for the formula.
In the denominator we write the total number of velocities given to us, which in our case is 6. Therefore, in the formula, we write:
${v_{rms}} = \sqrt {\dfrac{{108}}{6}} $
where the numerator is the sum of squares of all the velocities and the denominator is the total number of velocities (just like we do in taking the average).
The rms velocity comes to be:
$v_{rms} = \sqrt{18} = 3 \sqrt{2}$ = 4.2426 units.

So, the correct answer is “Option C”.

Additional Information:
The rms velocity is generally encountered when we study kinetic theory of gases. The velocities of the molecules in a gas form a Maxwellian distribution. The distribution decides the number of molecules having velocity in a particular range i.e., between v and v + dv (say).

Note:
The term root mean squared average is also used with alternating current supplies but it has nothing to do with the rms velocity that we found here. Though the definition and the concept are the same for calculations of both but in one, we perform integration and in other we perform summation.