Recall that your trip to Mars is accomplished by using an el

Recall that your trip to Mars is accomplished by using an elliptic transfer orbit going from Earth to Mars as shown in Fig. 1. This trajectory assumes that Earth at departure, the Sun, and Mars at arrival, are aligned. Given the semi-major axis and the eccentricity of this transfer orbit as 195600000 km and 0.2095 respectively, calculate the spacecraft interplanetary velocity with respect to the Sun when arriving near Mars? Give your answer in km/s.

The Answer is 21.064 km/s.

can you show me the steps how to calculate it? thanks

Solution

v² = GM [2 / r - 1 / a]

a = 195.6 * 10⁹ m

M = 2 x 10³⁰kg , G = 6.673 * 10^-11m³kg^-1s^-2

v² = 6.673 * 10^-11 x 2 x 10³⁰ x (2/226.9 - 1/195.6 ) x 10^-9

v = 22 km/s