Answer to Question #113796 in Physics for Atif

Question

10. A miror reflector was placed on the moon during the Apollo moon missions and can reflect laser pulses sent through telescopes on the Earth to help in determining the distance to the moon. Laser beams are used because they remain tightly focused for large distances. Nevertheless, there is enough dispersion of the beam that it is about 7 kilometers in diameter when it reaches the Moon and 20 kilometers in diameter when it returns to Earth. Because of this very weak signal, observations are made for several hours at a time. By averaging the signal for this period, the distance to the Moon can be measured to an accuracy of about 3 centimeters (the average distance from the Earth to the Moon is about 385,000 kilometers). Knowing this distance,
what is the roundtrip time of a laser pulse to the moon to 3 significant digits? (lncidentally
the moon is receding from the carth at a rate of 3.8 cm per year

Answer 1

The speed of light is $c=2.998\cdot 10^8m/s$ . Since the average distance from the Earth to the Moon is about $L=385,000km=3.85\cdot 10^8m$ for travel time we get

(1) $\Delta t=2\cdot L/c=\frac{7.7\cdot 10^8m}{2.998\cdot 10^8m/s}=2.568s$

In order to measure the distance with an accuracy of about $\delta L=3cm$ it is necessary to measure the time (1) with an accuracy of up to

(2) $\delta t=2\cdot \delta L/c=\frac{0.06m}{3\cdot 10^8m/s}=2.\cdot 10^{-10}s=0.2ns$

To compute the lunar distance precisely, many factors must be considered in addition to the round-trip time of about 2.5 seconds. These factors include the location of the Moon in the sky, the relative motion of Earth and the Moon, Earth's rotation, velocity of light in various parts of air, propagation delay through atmosphere, the location of the observing station and its motion [1].

[1] https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment