Problem Graphic

NASA is planning an unmanned mission to Mars. Mission planners need the precise position of the planet relative to Earth on the launch date in order to determine the proper velocity for the spacecraft. If the distance or angle is incorrect, the spacecraft could be lost.

What is known:

• The radius of Earth's orbit r_e is 150 x 10⁶ km.
• The radius of Mars's orbit r_m is 207 x 10⁶ km.
• The angle α between Earth and Mars relative to the Sun is 30°.
• Assume both Earth's and Mars' orbits around the Sun is circular.

What is the position of Mars relative to Earth on the launch day?

• Draw a diagram and label the key points and known position vectors.
• Determine the relationship between the unknown position vector and those that are given.
• Using geometry, solve for the length and angle of the unknown vector.