Answer:
The first choice: Mars travels at the highest speed at the closest approach to the sun.
Step-by-step explanation:
Assume that the sun is stationary and that the gravitational pull from the sun is the only force on Mars.
The mechanical energy of Mars is the sum of two components:
- The gravitational potential energy
of Mars in relation to the sun, and - The kinetic energy of planet Mars,
.
Let
denote the distance between the sun and Mars. The
between the two would be proportional to
. Thus, increasing the distance between the sun and Mars would increase the
of Mars in relation to the sun.
The kinetic energy
of Mars is proportional to the square of the speed of Mars. Increasing the speed of Mars would thus add to the
of this planet.
Since the gravitational pull from the sun is the only force on Mars, the mechanical energy of Mars would be conserved. In other words, the sum of the
of Mars and the
of Mars in relation to the sun would be constant.
Thus, as the
of Mars decrease (as the planet moves closer to the sun,) the
of Mars (and thus the speed of this planet) would necessarily increase.
The speed of Mars is maximized when the
of this planet is maximized. Because of the conservation of
, the
of Mars in relation to the sun would need to be minimized. Thus, the distance between Mars and the sun at that moment would also need to be minimized. Hence, Mars would be travelling at the highest speed at its closest approach to the sun.