Relative Uncertainty/Absolute Uncertainty?

Question

Hey guys!~

Need some help on my physics homework for uncertainties.

The question is:

A mass is measured to be m = (4.4±0.2)kg and its speed is (18±2)m/s. Find the kinetic energy of the mass with its absolute uncertainty.

okay.. so i got this equation:

Kinetic Energy

= 1/2mv^2

= 1/2 x (4.4±0.2) x (18±2)^2

I first began solving (18±2)^2. I used the formula Q = a^n = ∆Q/Q = |n| x ∆a/a

So basically i got:

(18^2± |2| x 2/18)

|2| x 2/18 as being |n| x ∆a/a

=324±2/9

So then i solved for the absolute uncertainty ∆Q = (|n| x ∆a/a) x Q

= 324±(2/9 x 18)

= (324±4)

= 1/2 x (4.4±0.2) x (324±4)

= 1/2 x 1425.6±[(0.2/4.4 + 4/324) x 1425.6]

= 1/2 x (1425.6±82.4)

= 712.8±82.4) J

I'm pretty sure i got 712.8 correct but the 82.4 absolute uncertainty just doesn't seem right... My other 2 friends got an absolute uncertainty of around 192.5 as opposed to my answer of 82.4

Can anyone tell me where i went wrong and/or guide me through the steps of solving this equation? plz~~~

Thank you !!!

Answer 1

Your writing m = 4.4±0.2 and v² = 324±2/9 is not correct, because 4.4 and 324 represent the measure of  quantities (they have dimensions)while 0.2 and 2/9 are purely numbers (dimensionless ratio). After the sign ± you must put the absolute uncertainty, not the relative uncertainty.

Usually one evaluates the uncertainty on a measurement separately from calculating the measurement itself.

(a) Calculting the kinetic energy:

E = (1/2)mv²  = 0.5×4.4×18² = 712.8 J

(b) Calculating the uncertainty:

Δm/m = 0.2/4.4 =4.5%

Δv/v = 2/18 = 11%

ΔE/E =  ÃŽÂ”m/m + 2 Δv/v = 4.5% + 22% = 26.5 %

ΔE = 26.5E/100 = 26.5×712.8/100 = 188.9 J

Final result: E = 712.8 ± 188.9 J

This high uncertainty is due to the imprecision of the measurement of the speed, which introduces an uncertainty up to 11%

Your friend result 192.5 J was also correct as he rounded off the uncertainty on measurement of mass to 5% in place of 4.5 %

Answer 2

It seems to me that the formula is only good for small error and may not give the correct result here -

Why not do it the simple way, and work out

Greatest possible KE  -  Least Possible KE

= (4.4 +.2)*(18 + 2)^2   - (4.4 - .2)*(18 - 2)^2