A spacecraft is in a circular orbit around Earth at an altitude of 410 km. Its orbital velocity is calculated to be 7.82 m/s. The transfer orbit between Earth and Mars has a semi-major axis of 1.51x1011 m. The eccentricity of this transfer orbit is 0.6312. A velocity adjustment during midcourse of the trip would require burning the thruster for 5.12 minutes. The total trip duration between Earth and Mars along this transfer orbit is 243.14 days.

1. Q#1:
A Titan IV rocket has put your spacecraft in circular orbit
around Earth at an altitude of 410 km. What is your orbital
velocity? Give your answer in m/s.
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.
Q#2
Your trip to Mars is accomplished by using an elliptic transfer
orbit going from Earth to Mars as shown in Fig. 1. This
trajectory assumes that Earth at departure, the Sun, and Mars at
arrival, are aligned. Also, we will assume that Earth's and Mars'
orbits are circular, with radiuses R = 147294000 km and
228568000 km, respectively.
What is, in meters, the semi-major axis, a, of this transfer orbit?
Hint: determine its radiuses at aphelion and perihelion.
NOTE: Please answer this question in an integer format without
rounding or including any units. PLEASE SAVE THIS
ANSWER.
Q#3
Recall that your trip to Mars is accomplished by using an
elliptic transfer orbit going from Earth to Mars as shown in Fig.
1. This trajectory assumes that Earth at departure, the Sun, and
Mars at arrival, are aligned. Also, we will assume that Earth's
and Mars' orbits are circular, with radiuses R = 147294000 km
and 228568000 km, respectively.
What is the eccentricity, e, of the orbit?
NOTE: Please answer this question in a standard notation, 4
digits after the decimal point without rounding or including any

2. units.
Q#4
Around midcourse, a velocity adjustment is performed to
eliminate the small errors introduced when departing from Earth
orbit. This adjustment is performed using one of the onboard
thrusters. At the location where the adjustment is made, the
velocity is V = 26,500 m/s and should be V = 27,100 m/s.
Knowing that the thruster used for the maneuver generates a
thrust F = 7,805 N, determine how long, in minutes, it should be
turned on to adjust the velocity. The mass of the spacecraft is
2,500 kg
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.
Q#5
Recall that your trip to Mars is accomplished by using an
elliptic transfer orbit going from Earth to Mars as shown in Fig.
1. This trajectory assumes that Earth at departure, the Sun, and
Mars at arrival, are aligned. You calculated that the semi-major
axis for this transfer orbit in Q.2. Please refer that value.
How long, in days, would the interplanetary trip last? Hint:
first, determine the period of the transfer orbit.
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units
Q#6
Recall that your trip to Mars is accomplished by using an
elliptic transfer orbit going from Earth to Mars as shown in Fig.
1. This trajectory assumes that Earth at departure, the Sun, and
Mars at arrival, are aligned. You calculated the semi-major axis
and the eccentricity for this transfer orbit in Q.2 and Q.3
respectively (a, e values).
What is the spacecraft interplanetary velocity (in km/s) with

3. respect to the Sun when arriving near Mars.
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.
Q#7
Like for the Mars Pathfinder mission, the entry and landing on
Mars use a combination of aerodynamic drag (during entry, the
spacecraft is protected by a heat shield), rockets, parachutes,
and inflated airbags. The last phase of the entry & landing
sequence is controlled by the on-board computer system. When
the altitude reaches a certain critical value, the spacecraft
velocity is V = 40 m/s. At this altitude, the airbags are inflated
and a solid rocket engine is turned on to slow down the
spacecraft prior to impact on the Martian soil.
Knowing that the thrust generated by the rocket engine is 5,105
N and that the propellant burns for 10 s before impact, what will
be the velocity at impact (in m/s). Assume that the spacecraft
drag (due to parachute inflated airbags) is constant and is 7500
N, and that the spacecraft mass is 1,990 kg. Also, the Martian
gravitational acceleration is equal to 3.7 m/s2.
Hint: to solve this problem, make sure to include all forces
acting on the spacecraft (weight, drag and thrust).
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.
Q#8
The spacecraft is designed to leave the surface of Mars with the
first stage of its propulsion system and be put into Martian
orbit. Then, the second stage is used to boost the spacecraft
from Martian orbit into an interplanetary trajectory and return
to Earth.
If the spacecraft is in Martian orbit at an altitude of 356 km,
what is the velocity (in km/s) required to escape the
gravitational attraction of Mars. Note that the velocity direction

4. and magnitude required to actually return to Earth may be
different. (please use following values for Radius of Mars, and
Mass of Mars to get correct answers;
Radius of Mars: 3.395E6
Mass of Mars: 6.392E23
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.
Q#9
As discussed in the previous question, the second stage of the
propulsion system is used to boost the spacecraft from Martian
orbit into an interplanetary trajectory and return to Earth.
Knowing that the thrust generated by the second stage is 2,550
N and that the rocket exhaust velocity is V = 556m/s, what is
the required burn rate of propellant. Give your answer in kg/s.
NOTE: Please answer this question in a standard notation, 4
digits after the decimal point without rounding or including any
units.
Q#10
As discussed in the previous question, the second stage of the
propulsion system is used to boost the spacecraft from Martian
orbit into an interplanetary trajectory and return to Earth. The
burn rate of propellant was calculated in the previous question
Q.9, and the thruster must be turned on for 2 minutes in order
to reach the proper velocity.
What is the minimum amount of propellant needed to have a
chance of returning to Earth? Give your answer in kg.
NOTE: Please answer this question in a standard notation, 2
digits after the decimal point without rounding or including any
units.