Question

You are planning a mission to Mars. You want to send a 3-ton spacecraft there (3 tons of wet mass, is the initial mass of the spacecraft). As all the engineers working for you are calling in sick, you will have to design the mission yourself. (Mars radius is 3'390km).

What is the arrival excess velocity (in km/s), when reaching Mars' sphere of influence (following A, you were on a Hohmann transfer trajectory)?

Answer 1

The arrival excess velocity when reaching Mars' sphere of influence is 5.76 km/s.

Step-by-step explanation:

Step 1: Calculate the total energy of the spacecraft at Earth's orbit**

The total energy of the spacecraft at Earth's orbit is the sum of its kinetic energy and potential energy. The kinetic energy is given by:

KE = 1/2 * mv^2

where:

• KE is the kinetic energy in joules (J)
• m is the mass of the spacecraft in kilograms (kg)
• v is the velocity of the spacecraft in meters per second (m/s)

The potential energy is given by:

PE = -GMm/r

where:

• PE is the potential energy in joules (J)
• G is the gravitational constant, which is 6.67430 × 10^-11 m^3 kg^-1 s^-2
• M is the mass of the Earth, which is 5.972 × 10^24 kg
• r is the distance between the spacecraft and the center of the Earth in meters (m)

At Earth's orbit, the velocity of the spacecraft is approximately 29.8 km/s. The Earth's radius is approximately 6,371 km. Substituting these values into the equations above, we get:

KE = 1/2 * 3000 kg * (29.8 km/s)^2 = 8.83 × 10^8 J

PE = -6.67430 × 10^-11 m^3 kg^-1 s^-2 * 5.972 × 10^24 kg * 6,371,000 m = -3.62 × 10^10 J

Therefore, the total energy of the spacecraft at Earth's orbit is:

E = KE + PE = 8.83 × 10^8 J - 3.62 × 10^10 J = -2.74 × 10^10 J

Step 2: Calculate the total energy of the spacecraft at Mars' sphere of influence

The total energy of the spacecraft at Mars' sphere of influence is also the sum of its kinetic energy and potential energy. However, the potential energy is negative, as the spacecraft is escaping Earth's gravity and entering Mars' sphere of influence. The radius of Mars' sphere of influence is approximately 57.6 million kilometers. Substituting the values into the equations above, we get:

KE = 1/2 * 3000 kg * v^2

PE = -6.67430 × 10^-11 m^3 kg^-1 s^-2 * 6.4171 × 10^23 kg * 57.6 × 10^6 m = -2.03 × 10^11 J

Therefore, the total energy of the spacecraft at Mars' sphere of influence is:

E = KE + PE

E = 1/2 * 3000 kg * v^2 - 2.03 × 10^11 J

Step 3: Solve for the arrival excess velocity

Setting the total energy of the spacecraft at Earth's orbit equal to the total energy of the spacecraft at Mars' sphere of influence, we get:

-2.74 × 10^10 J = 1/2 * 3000 kg * v^2 - 2.03 × 10^11 J

Solving for v, we get:

v ≈ 5.76 km/s

Therefore, the arrival excess velocity when reaching Mars' sphere of influence is 5.76 km/s.