Newtonian Mechanics — Question 7

Question 7

A classical prediction should not be extrapolated without limits

A supplied comparison gives a classical prediction $q_C=10.00$ and a more complete prediction $q_R=10.01$ at condition L. At condition H the predictions are 10.00 and 13.00. The dimensionless observable must be predicted within 1% of $q_R$.

1. Compute the relative discrepancy at each condition.

2. At which condition is the classical prediction adequate by this criterion?

3. Explain how this illustrates the limited domain of Newtonian mechanics.

Question 7 - Solution

A classical prediction should not be extrapolated without limits

(a) Use the specified denominator.

$$\epsilon_L=\frac{|10.00-10.01|}{10.01}=0.0999\%,\quad\epsilon_H=\frac{3}{13}=23.08\%$$ At L the discrepancy is $0.01/10.01\approx0.0999\%$. At H it is $3/13\approx23.08\%$.

(b) Compare with one percent.

$$\boxed{\epsilon_L<1\%<\epsilon_H}$$ L passes and H fails. The decision follows the supplied comparison, not the names assigned to the two models.

(c) Keep an effective theory within its domain.

$$D_{\mathrm{adequate}}=\left\{u:\frac{|q_C(u)-q_R(u)|}{|q_R(u)|}\le0.01\right\}$$ Newtonian mechanics is a powerful approximation for ordinary macroscopic motion under suitable conditions. Relativistic or quantum descriptions become necessary outside its domain. Failure in another regime does not erase its usefulness where its errors meet the required tolerance.