The Martian#
Day 1: Fluids and Gravitation#
Problem 1
During the opening storm on Mars, Mark Watney is struck by a piece of launch debris and badly injured. In the movie, the wind is strong enough to force the crew to abort the mission. Assume the debris is a flat rectangular panel with area \(A = 0.30\ {\rm m^2}\) and mass \(m = 8.0\ {\rm kg}\). Assume the Martian atmospheric density is \(\rho = 0.020\ {\rm kg/m^3}\). Model the drag force on the panel as
where \(C_D = 1.2\). Suppose the panel must experience a horizontal acceleration of at least \(a = 15\ {\rm m/s^2}\) in order to become dangerous debris. You may neglect friction with the ground and assume all of the drag force accelerates the panel horizontally.
(a) What drag force is required to produce this acceleration?
(b) What wind speed would be required to generate that drag force?
(c) Compare your result to a very strong terrestrial storm speed of \(40\ {\rm m/s}\). Based on your calculation, is the opening storm scene realistic?
Problem 2
After the airlock failure, Watney repairs the Hab by covering the damaged opening with a flexible patch. Later, the Hab catastrophically fails again. We want to estimate the force on the repaired section. Assume the repaired opening is approximately circular with radius \(r = 0.45\ {\rm m}\). The Hab interior is maintained at pressure \(P_{\rm in} = 101{,}000\ {\rm Pa}\), while the exterior Martian pressure is \(P_{\rm out} = 600\ {\rm Pa}\).
(a) What is the pressure difference across the patch?
(b) What is the area of the opening?
(c) What outward force does the air inside the Hab exert on the patch?
(d) Express that force as an equivalent weight in kilograms on Earth by dividing by \(g = 9.81\ {\rm m/s^2}\).
(e) Based on this calculation, explain why a temporary patch might not remain reliable over long periods, especially if the material is flexing, aging, or exposed to large temperature changes. Assume the pressure is uniform across the patched opening.
Problem 3
After Watney is found alive, the Hermes crew eventually uses a gravity-assist maneuver near Earth to return toward Mars instead of simply turning the spacecraft around. To see why gravity matters, compare the gravitational pull of Earth and the Sun on the Hermes spacecraft when it is near Earth.
Use the following values:
Mass of the Sun: \(M_{\odot} = 1.99\times 10^{30}\ {\rm kg}\)
Mass of Earth: \(M_E = 5.97\times 10^{24}\ {\rm kg}\)
Distance from the spacecraft to Earth’s center during the flyby: \(r_E = 4.2\times 10^7\ {\rm m}\)
Distance from the spacecraft to the Sun: \(r_{\odot} = 1.50\times 10^{11}\ {\rm m}\)
Universal gravitation constant: \(G = 6.67\times 10^{-11}\ {\rm N\,m^2/kg^2}\)
(a) Compute the gravitational field due to Earth at the spacecraft, \(g_E = GM_E/r_E^2\).
(b) Compute the gravitational field due to the Sun at the spacecraft, \(g_{\odot} = GM_{\odot}/r_{\odot}^2\).
(c) Which gravitational influence is larger at that moment?
(d) Explain why a spacecraft can use a close flyby of Earth to significantly change its trajectory even though the Sun dominates the overall structure of the Solar System. Your explanation should connect your calculation to the idea of a gravity assist.
Day 2: Launch, Atmospheres, and Momentum#
Problem 4
At the end of the movie, Watney launches from the Martian surface in the stripped-down Mars Ascent Vehicle (MAV) to rendezvous with Hermes in orbit. Assume the MAV must reach a circular orbit at altitude \(h = 250\ {\rm km}\) above the Martian surface. Neglect atmospheric drag and assume the orbit is circular.
Use the following values:
Mass of Mars: \(M_M = 6.42\times 10^{23}\ {\rm kg}\)
Radius of Mars: \(R_M = 3.39\times 10^6\ {\rm m}\)
Universal gravitation constant: \(G = 6.67\times 10^{-11}\ {\rm N\,m^2/kg^2}\)
(a) Find the orbital radius \(r = R_M + h\).
(b) Use \(v_{\rm orb} = \sqrt{\frac{GM_M}{r}}\) to calculate the orbital speed required.
(c) Compare this to the orbital speed near Earth’s surface, about \(7.9\times 10^3\ {\rm m/s}\). Why is launching from Mars easier than launching from Earth?
(d) Explain why reducing the mass of the MAV helps make the launch more feasible.
Problem 5
During the final launch, Watney does not use a heat shield. In class, we usually associate heat shields with spacecraft moving through atmospheres at high speed. Let us estimate whether the Martian atmosphere would produce a large drag force during ascent. Assume the MAV has a cross-sectional area of \(A = 12\ {\rm m^2}\) and drag coefficient \(C_D = 0.80\). Assume the atmospheric density near the launch site is \(\rho = 0.020\ {\rm kg/m^3}\) and the launch speed at one moment is \(v = 900\ {\rm m/s}\).
The drag force is modeled by \(F_D = \frac{1}{2} C_D \rho A v^2.\)
(a) Calculate the drag force on the MAV at this speed.
(b) Compare this to the weight of a \(10{,}000\ {\rm kg}\) vehicle on Mars, where \(g_M = 3.71\ {\rm m/s^2}\).
(c) Based on your result, explain why aerodynamic heating and drag during ascent from Mars are much less severe than atmospheric reentry at Earth.
(d) Why would a spacecraft returning to Earth need a heat shield even if a spacecraft launching upward from Mars does not? Assume the density remains constant for this estimate.
Problem 6
In the final rescue scene, Watney punctures his spacesuit and uses escaping air to propel himself toward Commander Lewis. We will estimate whether this could produce a useful change in speed. Assume the total mass of Watney plus suit is \(m = 140\ {\rm kg}\). Suppose a small amount of gas escapes backward with speed \(u = 300\ {\rm m/s}\) relative to Watney, and the total escaping gas mass is \(\Delta m = 0.080\ {\rm kg}\).
Use conservation of momentum to estimate Watney’s recoil speed. For this problem, approximate the speed change by
Assume no other forces act during the maneuver.
(a) Calculate Watney’s change in speed.
(b) If he continues drifting at this speed for \(20\ {\rm s}\), how far would he move?
(c) If the gap to Lewis were \(15\ {\rm m}\), would this maneuver be enough to matter?
(d) Based on your calculation, is the “Ironman” scene completely impossible, somewhat exaggerated, or reasonably plausible?