The average distance of the earth from the sun is about 1.5 * 10^8 km. Assume that the earth's orbit around the sun is circular and that the sun is at the origin of your coordinate system. (a) Estimate the speed of the earth as it moves in its orbit around the sun. Express your answer in miles per hour with the appropriate number of significant figures. (b) Estimate the angle θ between the position vector of the earth now and what it will be in 4 months. (c) Calculate the distance between these two positions.

Distance between sun and earth will act as radius of circular path

R = 1.5 X 10¹¹ m

angular velocity of the earth

ω = 2π / T

= 2 X 3.14 / (365 X 24 X 60 X 60)

= 1.99 X 10⁻⁷ radian / s

velocity v = ω R

= 1.99 X 10⁻⁷ X 1.5 X 10¹¹

= 2.985 X 10⁴ m / s

= 29.85 km/s

29.85 x 60 x 60 km/h

= 107460 km/h

107460/1.6 mile / h

= 67162.5 mile / h

b) Angle moved in 12 months

= 360 degree

angle moved in 4 months

= 360 / 3

= 120 degree

c) Circumference of the orbit

= 2 x 3.14 x 1.5 x 10⁸ km

9.42 x 10⁸ km

required distance will be 1/3 rd of circumference

= 9.42 / 3 x 10⁸ km

= 3.14 x 10⁸ km