Question

Question 2 A space agency plans to send a probe to the Sun by launching it into an elliptical orbit with apoapsis at the Earth, which orbits a distance ae = 150 x 10km from the Sun. The probe is designed to withstand temperatures at a periapsis equal to ten times the Sun's radius Rs, where Rs = 696 x 10'km. a) Calculate the eccentricity of the elliptical orbit. (3 marks) (b) Assume that the probe is placed into the correct orbit after escaping the Earth and when it is still at apoapsis. What is its speed relative to the Earth? Recall that Newton's universal constant of gravitation is G = 6.67 x 10-11m'kg-15-2 and that the mass of the Sun is 1.99 x 1030kg. (4 marks)

Answer 1

A). The point where the satellite is the farthest away is called apoapsis. The semi major axis is on the line segment between the periapsis and apoapsis. And it is the half of the distance between them. The apoapsis distance is given by, a(1+e)=150x100km (1) The periapsis distance is given by, a(1-e)=696x109km (10) =6.96x10 km The semi major axis given by adding both equations 2a=156.96x10 a= 78.48x10ʻkm The eccentricity is given by subtracting both equations. 2ae=143.04x10 143.04x100 2(78.48x109) = 0.9113 e=

(B) The velocity of the earth and probe is V= Gm Vag (6.67x10-11)(1.99x1030 kg) v ap (6.67x10-11 mºkg='s-2)(1.99x1030 kg) 150x10° x10 m = 29747.04 m/s The velocity of the probe relative to earth is VpE = 2 x(29747.04 m/s) = 59494m/s