If, instead of the correct formula for kinetic energy, we carelessly use the classical expression for kinetic energy for a particle moving at half the speed of light, by what magnitude percent will our calculation be in error? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$
A) $27 \%$
B) $7 \%$
C) $19 \%$
D) $24 \%$
E) $1 \%$

Question

Quizplus · Accepted Answer

The classical kinetic energy formula is $ KE = \frac{1}{2}mv^2 $, while the relativistic kinetic energy is $ KE = (\gamma - 1)mc^2 $, where $\gamma = \frac{1}{\sqrt{1 - (v^2/c^2)}}$. For $v = 0.5c$, $\gamma \approx 1.1547$. The classical calculation gives $0.125mc^2$, while the relativistic calculation gives $0.1547mc^2$. The error is $(0.1547 - 0.125)/0.1547 \approx 0.192$, or 19%.

If, Instead of the Correct Formula for Kinetic Energy, We (c=3.0×108 m/s)\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)(c=3.0×108 m/s)

If, Instead of the Correct Formula for Kinetic Energy, We $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$