Question

NASA going to launch a rocket to Mars, with the rocket blasting off from the Earth when the Earth and Mars are aligned along a straight line from the Sun. If the Mars is now 60° ahead of the Earth in its orbit around the sun, when should NASA launch the rocket? All the planets orbit the sun in the same direction and 1 year on Mars is 1.9 Earth-years. Assume circular orbits for the Earth and the Mars.

Answer 1

Apply$\theta=wt$

For alignment the earth must move through 60° more than mars in the same time t

$w_c=\frac{360°}{yr}$

$w_m=\frac{360°}{1.9yr}$

$\theta_c=\theta_m+60°$

$w_ct=w_mt+60°$

$w_c=\frac{60°}{w_c-w_m}$

$t=\frac{60°}{\frac{360}{1yr}-\frac{360°}{1.9yr}}$

$t=\frac{60°}{360}(\frac{1}{\frac{0.9yr}{1.9yr^2}})$

$t=0.352yr=128days$

Earth has a larger angular velocity then mass and completes one orbit in less time