A radar signal is beamed towards a planet and its echo is received 7 minutes later. If the distance between the planet and the earth is 6.3 x 10¹⁰ m, calculate the speed of the signal.

The correct option is 3x10⁸ ms⁻¹.

To understand why this is correct, we can break down the problem step-by-step using the basic physics of wave propagation.

Step 1: Understand the travel path of the radar signal
A radar signal is sent from Earth, travels to the planet, reflects off its surface (creating an echo), and then travels back to Earth.
Let $d$ be the distance between Earth and the planet.
The total distance traveled by the signal ($d_{\text{total}}$) during the round trip is twice the distance to the planet:

$d_{\text{total}}=2\times d$

Given that the distance $d=6.3\times {10}^{10}\text{m}$, we calculate the total distance:

$d_{\text{total}}=2\times 6.3\times {10}^{10}\text{m}=12.6\times {10}^{10}\text{m}$

Step 2: Convert the time into standard units (seconds)
The total round-trip time ($t$) is given as 7 minutes. Since standard speed calculations require time in seconds, we convert minutes to seconds:

$t=7\times 60\text{seconds}=420\text{s}$

Step 3: Calculate the speed of the signal
The speed ($v$) of the signal is the total distance traveled divided by the total round-trip time:

$v=\frac{d_{\text{total}}}{t}$

Substituting the values we have:

$v=\frac{12.6\times {10}^{10}}{420}$

We can simplify this calculation:

$v=\frac{12.6}{420}\times {10}^{10}$

$v=0.03\times {10}^{10}{\text{ms}}^{-1}$

Converting this to standard scientific notation:

$v=3\times {10}^8{\text{ms}}^{-1}$

This matches the speed of light in a vacuum, which is expected for radar (electromagnetic) signals. Therefore, the speed of the signal is 3x10⁸ ms⁻¹.