Question

A spacecraft is coasting toward Mars. The mass of Mars is 6.4 × 10²³ kg and its radius is 3400 km (3.4 × 10⁶ m). When the spacecraft is 8900 km (8.9 × 10⁶ m) from the center of Mars, the spacecraft's speed is 2250 m/s. Later, when the spacecraft is 5200 km (5.2 × 10⁶ m) from the center of Mars, what is its speed? Assume that the effects of Mars's two tiny moons, the other planets, and the Sun are negligible. Precision is required to land on Mars, so make an accurate calculation, not a rough, approximate calculation.

Answer 1

The spacecraft is coasting, meaning it is moving without using its engines, only under the influence of gravitational pull of Mars.

If we neglect effects of Mars's two tiny moons, the other planets, and the Sun, we can safely assume that the mechanical energy of a coasting spacecraft is conserved. Mechanical energy is the sum of kinetic and potential energies.

Let m be the mass of the spacecraft and M be the mass of Mars. The kinetic energy of the spacecraft when it is moving at speed v is K=mv²/2. When spacecraft is at distance r from the center of the Mars, its gravitational potential energy is U=-GMm/r.

At one instance:

and the spacecraft is from the center of Mars.

At the other instance:

and the spacecraft is from the center of Mars.

By the conservation of mechanical energy of the spacecraft, the mechanical energy at the two instances are equal

Substituting given values and using we get