Apollo 13#
Day 1: Momentum, Pressure, and Energy#
Problem 1
During launch, the Saturn V rocket accelerates the Apollo 13 spacecraft upward from rest. Assume that at one moment during ascent the spacecraft and attached stage have a total mass of \(m = 2.8\times 10^6\ {\rm kg}\) and are accelerating upward at \(a = 4.2\ {\rm m/s^2}\). Take \(g = 9.81\ {\rm m/s^2}\).
(a) What is the net upward force required to produce this acceleration?
(b) What thrust must the engines provide?
(c) Express the thrust as a multiple of the spacecraft weight. Assume the rocket is moving vertically and neglect air resistance at this instant.
Problem 2
After the oxygen tank explosion, the crew must conserve consumables and reduce heating loads. One issue is the force caused by pressure inside a spacecraft hatch. Suppose a circular hatch has radius \(r = 0.50\ {\rm m}\). The cabin is maintained at pressure \(P_{\rm in} = 34{,}000\ {\rm Pa}\), while space outside is a vacuum.
(a) What is the pressure difference across the hatch?
(b) What is the area of the hatch?
(c) What outward force does the cabin air exert on the hatch?
(d) Express this force as an equivalent weight in kilograms on Earth. Assume the pressure is uniform across the hatch.
Problem 3
To conserve electrical power after the accident, the spacecraft must avoid wasting energy. Suppose one heater draws \(P = 180\ {\rm W}\) and is accidentally left on for \(t = 9.0\ {\rm h}\). Use \(1\ {\rm kWh} = 3.60\times 10^6\ {\rm J}\).
(a) How much electrical energy does the heater use in joules?
(b) Convert this energy to kilowatt-hours.
(c) If the battery stores \(1.2\times 10^7\ {\rm J}\), what percentage of the battery energy would this heater consume?
Day 2: Gravitation and Reentry#
Problem 4
Apollo 13 uses the Moon’s gravity and a free-return trajectory to help bring the crew back to Earth. Assume the spacecraft passes the Moon at a distance \(r = 5.0\times 10^6\ {\rm m}\) from the Moon’s center. Assume the spacecraft can be treated as a point mass.
Use the following values:
Mass of the Moon: \(M_M = 7.35\times 10^{22}\ {\rm kg}\)
Universal gravitation constant: \(G = 6.67\times 10^{-11}\ {\rm N\,m^2/kg^2}\)
(a) Compute the gravitational field strength due to the Moon at the spacecraft.
(b) What is the gravitational force on a \(m = 4.5\times 10^4\ {\rm kg}\) spacecraft at this location?
(c) Explain why even a brief close approach to the Moon can significantly alter the spacecraft trajectory.
Problem 5
During reentry, the command module enters Earth’s atmosphere at very high speed. We can estimate its kinetic energy. Assume the command module has mass \(m = 5.8\times 10^3\ {\rm kg}\) and reentry speed \(v = 1.10\times 10^4\ {\rm m/s}\).
(a) What is the kinetic energy of the command module at this speed?
(b) If only \(2.0\%\) of this kinetic energy is transferred as thermal energy into the heat shield, how much energy is absorbed by the shield?
(c) Based on your result, explain why reentry is such a severe thermal problem.
Problem 6
After reentry, the command module descends under parachutes for splashdown. Suppose the capsule mass is \(m = 5.8\times 10^3\ {\rm kg}\) and it descends at a constant speed of \(v = 8.0\ {\rm m/s}\) just before hitting the ocean. Assume the capsule is moving vertically and neglect buoyancy in air.
(a) What is the weight of the capsule?
(b) If the capsule is descending at constant speed, what must the total upward drag force from the parachutes be?
(c) If three parachutes share the load equally, what drag force acts on each parachute?