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Average speed and RMS speed, given speeds of objects

Physics Asked on May 25, 2021

I am given the speeds of eight objects, and I have to find the average speed and the RMS speed of these objects.
I tried taking the actual average of these speeds: $frac{1}{8}Sigma(v_i)$, but I get $7.4$ $m/s$ instead of $7.5$.

I know that $v_{average}=2 sqrt{frac{2 k_B T}{pi m}}$ and $v_{rms}=sqrt{frac{3 k_B T}{m}}$, but I do not see how these formulas could help me in any way as I do not know the mass of the given objects, just their velocities and their number.

I also know that $f(v)=(frac{m}{2 pi k_B T})^{frac{3}{2}} 4pi v^2e^{frac{-mv^2}{2k_BT}}$ is the probability per unit speed. I feel like it might have some relevance in terms of weighting in both cases but again, I do not know the masses of the objects. I tried assuming they all had the same mass and did not get the result I wanted.

I tried to find the rms speed by taking $sqrt{(Sigma (vi-v_{average})^2)}$ but I get 11.2 instead of 8.5.

Any suggestions on the right direction?

NOTE: The objects’ velocities in m/s are 2,4,5,5,8,9,11,15 (a user asked previously. I can remove if too specific).

One Answer

The average speed is indeed the simple numerical mean of the given speeds

The RMS speed is the root of the mean of the squares i.e. it is the square root of the average of the squares of the given speeds.

For a simple example. If there are 3 objects with speeds 1,2 and 3.

Then average speed = (1+2+3)/3 = 2

rms speed = sqrt((1+4+9)/3) =sqrt(14/3) = 2.16

Correct answer by silverrahul on May 25, 2021

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