# Atomic Clock Problem: Calculate Nanoseconds Difference

• Tonia
In summary, the conversation discusses the calculation of the factor for time dilation in an atomic clock moving at 1,000 km/h for 1.00 h as measured by an identical clock on Earth. The correct equation is 1/√(1-v^2/c^2), where v is the speed of the clock and c is the speed of light. However, the individual in the conversation made several errors in their calculation, such as not converting km to m and not squaring c. Ultimately, the correct answer is supposed to be 1.54 nanoseconds.
Tonia

## Homework Statement

An atomic clock moves at 1,000 km/h for 1.00 h as measured by an identical clock on the Earth. At the end of the 1.00 h interval, how many nanseconds slow will the moving clock be compared with the Earth clock?

## Homework Equations

(1/ sqr rt 1 - v^2/c^2)

## The Attempt at a Solution

1,000 km/h = 278 m/s
Convert the 1,000 km/h to met/sec. first
(1/ sqr rt 1 - v^2/c^2)

= (1/ sqr rt 1 - 278^2/3.00 X 10^8 m/s)

I am not sure how to convert from 1,000 km/h to m/s, I think I divide it by 3600 seconds which would equal 0.277 m/s, so how do I change that to 278 m/s?

Tonia said:
I am not sure how to convert from 1,000 km/h to m/s, I think I divide it by 3600 seconds which would equal 0.277 km/s, so how do I change that to 278 m/s?
Convert also km-s to meters.

1,000 km/hr to met./sec. = 1,000km X 1,000met. = 1,000,000 met./ hr
1,000,000 met. divided by 3600 seconds = 277.7met./sec. = 278 met./sec.

Equation used: 1/ (squr rt 1 - v^2) = 1/ (squrrt 1 - 278^2/3.00 X 10^8 met./sec. = 1/ 1 - 77284/3.0 E8 = 1/-77283/3.0 E8 = 1/-2.5761e12 The Answer is supposed to be 1.54 Nanoseconds, what am I doing wrong?

You
Tonia said:
1,000 km/hr to met./sec. = 1,000km X 1,000met. = 1,000,000 met./ hr
1,000,000 met. divided by 3600 seconds = 277.7met./sec. = 278 met./sec.

Equation used: 1/ (squr rt 1 - v^2) = 1/ (squrrt 1 - 278^2/3.00 X 10^8 met./sec. = 1/ 1 - 77284/3.0 E8 = 1/-77283/3.0 E8 = 1/-2.5761e12 The Answer is supposed to be 1.54 Nanoseconds, what am I doing wrong?

You did not write any equation. As I see, you intended to calculate the factor ##\sqrt{1-\frac{v^2}{c^2}}##, but you forgot to square c, did not apply parentheses and made some more errors.
Look after Lorentz transformation in your Lecture Notes.

Multiply with conversion factors do get the values in the correct units eg. 1km = 1000 m
So you can multiply with the conversion factor 1 km / 1000 m and the value will not be
altered since you are multiplying by one . So 35.7 km = 35.7 km x 1000 m / 1 km = 35 700 m
The km units cancel each other out so only the meter units are leftover, which is the units

I figured out the problem but thanks.

## 1. What is an atomic clock?

An atomic clock is a highly accurate timekeeping device that uses the resonant frequency of atoms to measure time. It is considered the most accurate type of clock currently available, with a margin of error of only one nanosecond every 100 million years.

## 2. How does an atomic clock work?

An atomic clock uses the oscillations of atoms, usually cesium or rubidium, to measure time. The atoms are excited to a higher energy state and then allowed to fall back to their ground state, emitting electromagnetic radiation at a specific frequency. This frequency is used to keep time and is extremely stable and consistent.

## 3. What is the atomic clock problem?

The atomic clock problem refers to the challenge of accurately calculating the difference in time, measured in nanoseconds, between two atomic clocks. This problem is important in fields such as space navigation and telecommunications, where precise timing is crucial.

## 4. How do scientists calculate nanoseconds difference between atomic clocks?

To calculate the difference in nanoseconds between atomic clocks, scientists use a technique called time transfer. This involves comparing the time signals from two atomic clocks and using a series of calibration steps to account for any delays or errors in the measurement process. This results in a highly accurate calculation of the nanoseconds difference between the two clocks.

## 5. Why is accurately calculating nanoseconds difference important?

Accurately calculating nanoseconds difference between atomic clocks is important because it allows for precise synchronization of time in various systems and devices. This is crucial in fields such as GPS navigation, where even a small difference in time can lead to significant errors in location. It also enables more efficient and reliable communication between systems that rely on precise timing.

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