You have been presented with a source of electrons, ^14 C, having kinetic energy equal to 0.305

step 1 As we all know, that rest mass of electron is given by E0 is equal to MEC square. Also, I can wr

You have been presented with a source of electrons, ^14 C,...

You have been presented with a source of electrons, ${ }^{14} \mathrm{C}$, having kinetic energy equal to 0.305 times the rest energy. Suppose you have a pair of detectors that can detect the passage of the electrons without disturbing them. You wish to show that the relativistic expression for momentum is correct and the nonrelativistic expression is incorrect. If a $2.0-\mathrm{m}$ -long baseline between your detectors is used, what timing accuracy is needed to show that the relativistic expression for momentum is correct?

Key Concepts

Time-of-Flight Measurement and Timing Accuracy

Time-of-flight measurements involve determining the speed of a particle by measuring the time it takes to traverse a known distance. In experiments aimed at distinguishing relativistic from nonrelativistic predictions, the difference in the expected flight times (often on the order of nanoseconds) is critical. Achieving a timing accuracy on the order of a nanosecond (or a few nanoseconds) is necessary to resolve the small discrepancies between the two theoretical predictions, thereby confirming the relativistic expression for momentum.

Lorentz Factor

The Lorentz factor, ? = 1/?(1 - (v/c)²), appears in many relativistic formulas, including both kinetic energy and momentum. It quantifies the deviation from Newtonian mechanics as the speed v of the particle approaches the speed of light c, and it is essential for calculating the correct values when relativistic effects are significant.

Relativistic Kinetic Energy

Relativistic kinetic energy is expressed as K = (? - 1)mc², contrasting with the classical expression K = (1/2)mv². When a particle’s kinetic energy is a sizable portion of its rest energy, the relativistic formula must be used to accurately describe its energy content, which in turn affects its momentum and velocity calculations.

Relativistic Momentum

Relativistic momentum is defined as p = ?mv, where ? (the Lorentz factor) accounts for the effects of time dilation at high speeds. This concept is crucial when the particle’s kinetic energy is a significant fraction of its rest energy, as it leads to deviations from the classical momentum formula. The relativistic expression more accurately predicts the particle’s behavior, which becomes important in experiments involving high-speed particles.

Nonrelativistic Momentum

Nonrelativistic momentum, given simply as p = mv, assumes that the speeds involved are much less than the speed of light so that any relativistic effects are negligible. When the kinetic energy is not very small compared to the rest energy, this simple expression fails to account for the mass–velocity relationship dictated by special relativity, leading to noticeable discrepancies between predicted and actual behavior.

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