Question

Venus's average distance from the sun is 0.72 AU and Saturn's is 9.54 AU. Calculate the orbital velocity of Venus and Saturn around the sun. (The mass of the sun is 1.99 X 10^30 KG. an AU is 1.50 X 10^11 M.

Answer 1

So you can compute the force of gravity on the planet GMm/r^2
or (6.674 X 10^−11)(1.99 X 10^30)m / d^2(1.5 X 10^11)^2
Where d is the average distance of the planet form the sun and m is the mass of the planet (I will keep these as symbols so doing Saturn and Venus will be simple substitution into one formula in the end).

Now realize that if they are in stable circular orbit, then this force must provide the necessary centripetal force mv^2/r or mv^2 / d(1.5 X 10^11)

So we get:

(6.674 X 10^−11)(1.99 X 10^30)m / d^2(1.5 X 10^11)^2 = mv^2 / d(1.5 X 10^11)
The m's cancel out as does one 1/d(1.5 X 10^11)
(6.674 X 10^−11)(1.99 X 10^30) / d(1.5 X 10^11) = v^2
Evaluating and square rooting yeilds:
v = sqrt((8.854 X 10^8)/d)

Now plug in 0.72 for venus and 9.54 for saturn

Venus = 35067.39 m/s
Saturn = 9633.75 m/s

Some precision was lost rounding to 8.854 X 10^8, so if more presion is required just type the whole thing from before into a calculator.  Anyways the process is sound