(a) The distance the Earth travels every year is the perimeter of the orbit; since we have the average radius of the orbit, we can calculate the perimeter:
[tex]d=2\pi r = 2 \pi (1.5 \cdot 10^{11}m)=9.4 \cdot 10^{11}m[/tex]
(b) The average centripetal acceleration is given by
[tex]a_c = \frac{v^2}{r} [/tex]
where v is the average speed of the Earth, that we can find by dividing the distance covered by the Earth in one year (=the perimeter of the orbit) by the the time corresponding to 1 year:
t=1 year=365.25 days=3.2 \cdot 10^7 s
So the velocity is:
[tex]v= \frac{d}{t}= \frac{9.4 \cdot 10^{11}m}{3.2 \cdot 10^7 s} =2.94 \cdot 10^4 m/s[/tex]
And so the centripetal acceleration is:
[tex]a_c = \frac{v^2}{r}= \frac{(2.94 \cdot 10^4 m/s)^2}{9.4 \cdot 10^{11}m} =9.2 \cdot 10^{-4} m/s^2[/tex]
