Question

Neutrinos are subatomic particles with a very low mass. Recent work at CERN, Europe's particle-physics lab near Geneva, Switzerland, suggests that neutrinos may have the ability to travel faster than the speed of light. If the speed of light is \(1.86 \times 10^{8} \mathrm{mi} / \mathrm{hr}\), how many nanoseconds should it take for light to travel from CERN to the Gran Sasso National Lab in Italy, a \(730.0 \mathrm{~km}\) journey? If a neutrino can travel the same distance 60 nsec faster, how many significant figures would you need to detect the difference in speed?

Short answer

It takes light approximately \(8.78 \times 10^{6}\) nanoseconds to travel from CERN to Gran Sasso. A minimum of 5 significant figures is needed to detect the difference in speed of neutrinos.

Step-by-step solution

Convert distance from kilometers to miles

The distance from CERN to Gran Sasso National Lab is given as 730.0 km. Use the conversion factor to convert kilometers to miles. 1 km = 0.621371 miles, so: \[ \text{Distance in miles} = 730.0 \text{ km} \times 0.621371 \text{ miles/km} = 453.60183 \text{ miles} \]

Calculate time taken by light

Using the speed of light, calculate the time it would take for light to travel from CERN to Gran Sasso. The speed of light is \(1.86 \times 10^{8} \mathrm{mi/hr}\).\[ \text{Time}_\text{light} = \frac{\text{Distance}}{\text{Speed}} = \frac{453.60183 \text{ miles}}{1.86 \times 10^{8} \text{ miles/hr}} = 2.439 \times 10^{-6} \text{ hours} \]

Convert time from hours to nanoseconds

Convert the time from hours to nanoseconds. There are \(3.6 \times 10^{12}\) nanoseconds in an hour.\[ \text{Time}_\text{light} = 2.439 \times 10^{-6} \text{ hours} \times 3.6 \times 10^{12} \text{ nanoseconds/hour} = 8.78 \times 10^{6} \text{ nanoseconds} \]

Calculate time taken by neutrino

According to the problem, a neutrino can travel the same distance 60 nanoseconds faster than light.\[ \text{Time}_\text{neutrino} = 8.78 \times 10^{6} \text{ nanoseconds} - 60 \text{ nanoseconds} = 8.77994 \times 10^{6} \text{ nanoseconds} \]

Detecting the difference in speed

To detect the speed difference, calculate the proportion of the difference to the total time in significant figures.The time difference is 60 nanoseconds. The significant figures required to detect the speed difference can be approximately calculated as:\[ \text{Required significant figures} = \log_{10}\left(\frac{8.78 \times 10^{6}}{60}\right) \approx \log_{10}(1.463 \times 10^{5}) \approx 5 \]

Key concepts

Neutrinos

Neutrinos are fascinating subatomic particles. They are incredibly light and elusive, barely interacting with matter. Despite their small size and mass, they play a crucial role in the universe. Scientists study neutrinos to learn more about the fundamental particles and forces that make up everything we see. Recent experiments suggest that neutrinos might travel faster than the speed of light, challenging our understanding of physics as per Einstein's theory of relativity.

Significant Figures

Significant figures represent the precision of a measurement. Math and science calculations depend on them to ensure accuracy. For instance, in the exercise, converting kilometers to miles and calculating time involve significant figures to maintain precision. When working with very large or very small numbers, significant figures help communicate the reliability of the data. In the speed difference calculation in this problem, up to five significant figures were needed to detect the difference in speed accurately.

Conversion Factors

Conversion factors are used to convert a quantity from one unit to another. They are essential in calculations to maintain accuracy and consistency. In the exercise, kilometers were converted to miles using the factor 1 km = 0.621371 miles. This helps in understanding distances in familiar units and ensuring that other calculations, such as time, remain consistent. Conversion factors are used in various fields to standardize measurements and facilitate communication.

Subatomic Particles

Subatomic particles are the building blocks of atoms. They include protons, neutrons, and electrons, along with other particles like neutrinos and quarks. Understanding these particles is essential for the fields of particle physics and cosmology. Neutrinos, being one of the subatomic particles, interact very little with other matter, making them hard to detect but vital for understanding the universe's forces and elements. Their study helps physicists explore the limits of the Standard Model of particle physics.

Speed Calculation

Speed calculation often involves determining how far something travels over a certain period. The exercise involved calculating the speed of light and comparing it to neutrinos. The formula used was speed = distance / time. By converting units and carefully managing significant figures, the calculation showed how tiny differences in speed could be significant over large distances. Speed calculations are common in physics to understand motion, energy, and the behavior of various particles and objects in space and time.