Question

As its closest approach, the distance between Mars and the Earth is found to be \[60\,{\text{million km}}\]. When the two planets are at the closest, how long would it take to send a radio message from a space probe of Mars or Earth?
A. \[5\,{\text{s}}\]
B. \[200\,{\text{s}}\]
C. \[0.2\,{\text{s}}\]
D. \[20\,{\text{s}}\]

Answer 1

Hint:: Use the distance-speed formula. This formula gives the relation between speed, total distance travelled and time.

Formula used:
The expression for the speed of an object is
\[V = \dfrac{d}{t}\] …… (1)
Here, \[V\] is the speed of the object, \[d\] is the total distance travelled by the object and \[t\] is the time required to travel the same distance.

Complete step by step answer:
The closer distance between Mars and the Earth is \[60\,{\text{million km}}\].
The radio waves are sent from a space probe of Mars to the Earth.
The speed of light \[c\] is \[3 \times {10^8}\,{\text{m/s}}\]. The radio waves travel with the speed of light.
\[\Rightarrow c = 3 \times {10^8}\,{\text{m/s}}\]
When Mars and the Earth are at their closest approach, the distance between the space probe of Mars and the Earth is equal to the distance between Mars and the Earth.
Convert the distance \[d\] between the space probe of Mars and the Earth into SI system of units.
\[\Rightarrow d = \left( {60\,{\text{million km}}} \right)\left( {\dfrac{{{{10}^6}}}{{1\,{\text{million}}}}} \right)\left( {\dfrac{{{{10}^3}\,{\text{m}}}}{{1\,{\text{km}}}}} \right)\]
\[ \Rightarrow d = 6 \times {10^{10}}\,{\text{m}}\]
Rewrite equation (1) for the speed \[c\] of the radio waves.
\[\Rightarrow c = \dfrac{d}{t}\]
Rearrange the above equation for the time \[t\] required for the radio message from the space probe of Mars to reach the Earth.
\[\Rightarrow t = \dfrac{d}{c}\]
Substitute \[6 \times {10^{10}}\,{\text{m}}\] for \[d\] and \[3 \times {10^8}\,{\text{m/s}}\] for \[c\] in the above equation.
\[\Rightarrow t = \dfrac{{6 \times {{10}^{10}}\,{\text{m}}}}{{3 \times {{10}^8}\,{\text{m/s}}}}\]
\[ \Rightarrow t = 200\,{\text{s}}\]
Therefore, the time required for the radio message from the space probe of Maars to reach the Earth is \[200\,{\text{s}}\].
Hence, the correct option is B.

Note: Convert the unit of closest distance of approach between Mars and the Earth for the correct answer. The radio waves travel with the speed of light.