Thuyết tương đối, phương pháp thực nghiệm và các chủ đề chuyên sâu

Kinetic energy is not ½mv²

• At $v = 0.6c$, $\tfrac12 mv^2 = 0.18mc^2$. The relativistic kinetic energy is $(\gamma - 1)mc^2 = 0.25mc^2$ — 40% larger.
• GRE "specialised topics" mix relativity with laboratory skills: uncertainties, precision 精密度 and accuracy 准确度, and log plots.

Relativistic energy

• $\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$. Total energy $E = \gamma mc^2$; kinetic $K = (\gamma - 1)mc^2$; momentum $p = \gamma mv$.
• $E^2 = (pc)^2 + (mc^2)^2$. At $0.6c$: $\gamma = 1.25$, $p = 0.75\,mc$.

Combining uncertainties

• Independent errors add in quadrature 方和根. For a product or quotient, add relative uncertainties in quadrature: 3% and 4% give 5%.
• For $y = x^n$, the relative uncertainty is multiplied by $|n|$. Linear addition gives a worst-case bound, not the standard uncertainty.

Precision and accuracy

• Random scatter limits precision; calibration bias limits accuracy. Averaging shrinks random error by $1/\sqrt N$ but leaves bias unchanged.
• Counting $N$ random events: uncertainty $\sqrt N$.

Log plots

• If $y = Ax^n$, then $\log y = \log A + n\log x$: a straight line of slope $n$ on log–log axes.
• If $y = Ae^{kx}$, a semi-log plot ($\log y$ against $x$) is straight with slope $k$.

Recap