A particle is moving at $0.86 c$. By what magnitude percentage is the Newtonian expression for momentum in error? (The percentage error is the difference between the erroneous and correct expressions, relative the correct one).
A) $49 \%$
B) $62 \%$
C) $55 \%$
D) $40 \%$

Question

Quizplus · Accepted Answer

The correct expression for momentum in relativistic mechanics is $p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m$ is the mass of the particle, $v$ is its velocity, and $c$ is the speed of light. The Newtonian (classical) expression for momentum is simply $p = mv$. The percentage error is calculated as $\frac{p_{relativistic} - p_{Newtonian}}{p_{relativistic}} \times 100\%$.Given $v = 0.86c$, the relativistic factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - (0.86)^2}} \approx 2$. Thus, the relativistic momentum is about twice the Newtonian momentum, leading to a $100 \times (2 - 1) / 2 = 50\%$ error. The closest option provided is $49\%$, indicating that the Newtonian expression underestimates the momentum by approximately 49% compared to the relativistic expression.

A Particle Is Moving At 0.86c0.86 c0.86c by What Magnitude Percentage Is the Newtonian Expression for Magnitude

A Particle Is Moving At $0.86 c$ by What Magnitude Percentage Is the Newtonian Expression for Magnitude