Conversations with astronauts on the lunar surface were characterized by a kind of echo in which the earthbound person’s voice was so loud in the astronaut’s space helmet that it was picked up by the astronaut’s microphone and transmitted back to Earth. It is reasonable to assume that the echo time equals the time necessary for the radio wave to travel from the Earth to the Moon and back (that is, neglecting any time delays in the electronic equipment). Calculate the distance from Earth to the Moon given that the echo time was 2.56 s and that radio waves travel at the speed of light ( 3 x 10⁸m/s ).

Given and known data:

The echo time is t = 2.56 sec.

The speed of radio wave travel = \(c = 3 \times {10^8}\;{\rm{m}}/{\rm{s}}\).

The echo time is the time lag between the transmitting and receiving ends. This duration increases with the distance between the sender and the receiver.

The distance between the Earth and the Moon is given by

d = c.t

Here, d is the distance between the Earth and the Moon, c is the speed of light, and t is the echo time.

Substitute all the values in the above equation.

\(\begin{array}{l}d = \left( {3 \times {{10}^8}\;{\rm{m}}/{\rm{s}}} \right)\left( {2.56\;{\rm{s}}} \right)\\d = 7.68 \times {10^8}\;{\rm{m}}\end{array}\)

Therefore, the distance from Earth to the Moon for echo time is \(7.68 \times {10^8}\;{\rm{m}}\).