Based on the available information, the answers are 10.94 km/s, 1.18 million seconds, 1.02 km/s, and approximately 180°.
To calculate the various parameters for the spacecraft's journey from Earth to Mars, consider the orbital mechanics involved.
Here are the calculations for each part:
(a) The burnout speed relative to the Earth:
The burnout speed is the velocity the spacecraft needs to achieve to escape Earth's gravitational pull. Calculate it using the vis-viva equation:
v_burnout = √(2 * μ / r_initial - μ / a_transfer)
Where:
μ = gravitational parameter of Earth (3.986 * 10^14 m^3/s^2)
r_initial = initial radius of the spacecraft's circular orbit (300 km + radius of Earth)
a_transfer = semi-major axis of the transfer ellipse (r_initial + radius of Mars)
Using the given values:
r_initial = 300 km + 6,371 km (radius of Earth) = 6,671 km
a_transfer = 6,671 km + 3,389 km (radius of Mars) = 10,060 km
v_burnout = √(2 * 3.986 * 10^14 / 6,671,000 - 3.986 * 10^14 / 10,060,000)
v_burnout ≈ 10.94 km/s
Therefore, the burnout speed relative to the Earth is approximately 10.94 km/s.
(b) The time it takes the probe to reach the Moon's orbit on its way to Mars:
To calculate the time, determine the transfer time for the Hohmann transfer orbit. The transfer time can be calculated using the following equation:
t_transfer = π * √((a_transfer^3) / μ)
Using the given values:
t_transfer = π * √((10,060,000^3) / 3.986 * 10^14)
t_transfer ≈ 1.18 million seconds
Therefore, it takes approximately 1.18 million seconds for the probe to reach the Moon's orbit on its way to Mars.
(c) The speed of the probe relative to the Earth when it crosses the Moon's orbit:
The speed of the probe relative to the Earth when it crosses the Moon's orbit can be calculated using the vis-viva equation:
v_probe = √(μ * (2 / r_probe - 1 / a_transfer))
Where:
r_probe = radius of the Moon's orbit (3,844 km + radius of Earth)
Using the given values:
r_probe = 3,844 km + 6,371 km (radius of Earth) = 10,215 km
v_probe = √(3.986 * 10^14 * (2 / 10,215,000 - 1 / 10,060,000))
v_probe ≈ 1.02 km/s
Therefore, the speed of the probe relative to the Earth when it crosses the Moon's orbit is approximately 1.02 km/s.
(d) The angle the Moon moves through in the time it takes for the probe to coast to lunar orbit:
The angle the Moon moves through can be calculated using the formula:
θ = (t_transfer / T_Moon) * 360°
Where:
T_Moon = period of the Moon's orbit (27.3 days)
Using the given values:
T_Moon = 27.3 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute
T_Moon ≈ 2,360,800 seconds
θ = (1.18 million seconds / 2,360,800 seconds) * 360°
θ ≈ 180°
Therefore, the angle the Moon moves through in the time it takes for the probe to coast to lunar orbit is approximately 180°.