The distance between earth and the moon can be determined from the time it takes for a laser beam to travel from earth to a reflector on the moon and back. If the round-trip time can be measured to an accuracy of 0.19 of a nanosecond (1 ns = 10-9 s), what is the corresponding error in the earth-moon distance?
To determine the corresponding error in the Earth-Moon distance, we need to first understand the relationship between the round-trip time of a laser beam and the distance it travels.
The round-trip time of a laser beam is the time it takes for the beam to travel from Earth to the Moon and back. Since the laser beam travels at the speed of light, we can use the formula:
Distance = Speed × Time
In this case, the "Speed" is the speed of light, which is approximately equal to 3 × 10^8 meters per second. And the "Time" is the round-trip time of the laser beam.
Let's calculate the round-trip time first. Given that the time can be measured with an accuracy of 0.19 nanoseconds, we can convert it to seconds:
Accuracy in seconds = 0.19 ns × 10^(-9) s/ns
≈ 1.9 × 10^(-10) s
Now let's calculate the round-trip time including the accuracy:
Round-Trip Time = Measured Time + Accuracy
= Measured Time + 1.9 × 10^(-10) s
To find the error in the Earth-Moon distance, we need to calculate the distance using the round-trip time including the accuracy, and then subtract the distance calculated using the measured time.
Using the formula Distance = Speed × Time, we can calculate the Earth-Moon distance using the measured round-trip time:
Distance (measured) = Speed × Measured Time
And using the round-trip time including the accuracy, the Earth-Moon distance becomes:
Distance (with accuracy) = Speed × (Measured Time + 1.9 × 10^(-10) s)
Now, the corresponding error in the Earth-Moon distance can be calculated by subtracting the distance using the measured time from the distance with accuracy:
Error = Distance (with accuracy) - Distance (measured)
Hope this helps!