Taking the age of Earth to be about 4×10⁹ years and assuming its orbital radius of 1.5 ×10¹¹ m has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun).

Solution:

First, we need to compute for the linear velocity of the Earth using the formula below knowing that the Earth has 1 full revolution in 1 year

v=r\omega

where r=1.5\times 10^{11} \ \text{m} and \omega = 2\pi \ \text{rad/year} . Substituting these values, we have

\begin{align*}
v & = r \omega \\ \\
v & = \left( 1.5\times 10^{11} \ \text{m} \right)\left( 2 \pi \ \text{rad/year} \right) \\ \\
v & = 9.4248\times 10^{11} \ \text{m/year}
\end{align*}

Knowing the linear velocity, we can compute for the total distance using the formula

\Delta x = v \Delta t

We can now substitute the given values: v = 9.4248\times 10^{11} \ \text{m/year} and \Delta t = 4\times 10^{9} \ \text{years} .

\begin{align*}
\Delta x & = v \Delta t \\ \\
\Delta x & = \left( 9.4248\times 10^{11} \ \text{m/year}  \right) \left( 4\times 10^{9} \ \text{years} \right) \\ \\
\Delta x & = 3.7699 \times 10^{21} \ \text{m} \\ \\
\Delta x & = 4 \times 10^{21} \ \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}