Question

Radio waves of wavelength 6.00 ×${\mathbf{10}}^{\mathbf{2}}$m can be used to communicate with spacecraft over large distances.

(a) Calculate the frequency of these radio waves.

(b) Suppose a radio message is sent home by astronauts in a spaceship approaching Mars at a distance of 8.0 × ${\mathbf{10}}^{\mathbf{10}}$ m from earth. How long (in minutes) will it take for the message to travel from the spaceship to earth?

Short answer

1. The frequency of radio waves is $0.5\times {10}^6{s}^{-1}$.
2. The time taken to travel the message from spaceship to earth is $2.93\times {10}^{2}s$.

Step-by-step solution

Frequency

Frequency is thenumber of cycles/waves produced in one second (1s); more waves passing in one-second result in more frequency and vice versa.

Frequency of radio wave

The formula to calculate the wavelength of the radio wave is as follows:

$\text{Frequency}\left(v\right)=\frac{Speedoflight\left(c\right)}{wavelenght\left(\lambda \right)}$

Here:

• The speed of light is$3\times {10}^{8}m/s$.
• The wavelength of is$6.00\times {10}^{2}m/s$.

Substituting the known value in the formula ofthe speed of the wave is done as follows:

$\begin{array}{rcl}v& =& \frac{3\times {10}^{8}m/s}{6.00\times {10}^{2}m}\\ & =& 0.5\times {10}^6{s}^{-1}\end{array}$

Calculation of time

The formula to calculate the time takento travel the message from the spaceship to earth is as follows:

$Time=\frac{Dis\tan ce\left(d\right)}{Speedoflight\left(c\right)}$

Here:

• The speed of light is$3\times {10}^{8}m/s$.
• The distance of mars from earth is $8.8\times {10}^{10}m$.

Substituting the known value in the time taken to travel the messageis done as follows:

$$\begin{gathered} \text{Time}=\frac{8.8\times {10}^{10}m}{3\times {10}^{8}m/s} \\ =2.93\times {10}^{2}m \end{gathered}$$