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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Chapter 2 • Chapter Review 227

CHAPTER 2 PROJECT The Apollo Lunar Module
This Project may be done 2. What is the reference acceleration a ref (t)?
Michael Collins, Apollo 11, NASA The Lunar Module (LM) was a 4. reference jerk J ref (t).
individually or as part of a team.
The rate of change of acceleration is called jerk. Find the
3.
small spacecraft that detached from
The rate of change of jerk is called snap. Find the reference
the Apollo Command Module and
snap S ref (t).
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was designed to land on the Moon.
Evaluate r ref (t), v ref (t), a ref (t), J ref (t), and S ref (t) when t = 0.
5.
Fast and accurate computations
were needed to bring the LM from an orbiting speed of about But small variations in propulsion, mass, and countless other
5500 ft/s to a speed slow enough to land it within a few feet of a variables cause the LM to deviate from the predetermined path. To
designated target on the Moon’s surface. The LM carried a 70-lb correct the LM’s position and velocity, NASA engineers apply a
computer to assist in guiding it successfully to its target. The force to the LM using rocket thrusters. That is, they changed the
approach to the target was split into three phases, each of which acceleration. (Remember Newton’s second law, F = ma.)
followed a reference trajectory specified by NASA engineers. ∗ Engineers modeled the actual trajectory of the LM by
The position and velocity of the LM were monitored by sensors that
tracked its deviation from the preassigned path at each moment. 1 2 1 3 1 4
Whenever the LM strayed from the reference trajectory, control r(t) = R T + V T t + 2 A T t + 6 J A t + 24 S A t (2)
thrusters were fired to reposition it. In other words, the LM’s
position and velocity were adjusted by changing its acceleration. We know the target parameters for position, velocity, and
The reference trajectory for each phase was specified by the acceleration. We need to find the actual parameters for jerk and
engineers to have the form snap to know the proper force (acceleration) to apply.
1 2 1 3 1 4 6. Find the actual velocity v = v(t) of the LM.
r ref (t) = R T + V T t + A T t + J T t + S T t (1)
2 6 24 7. Find the actual acceleration a = a(t) of the LM.
The reference trajectory given in equation (1) is a fourth-degree 8. Use equation (2) and the actual velocity found in Problem 6 to
polynomial, the lowest degree polynomial that has enough free express J A and S A in terms of R T , V T , A T , r(t), and v(t).
parameters to satisfy all the mission criteria. Now we see that the 9. Use the results of Problems 7 and 8 to express the actual
parameters R T =r ref (0), V T = v ref (0), A T = a ref (0), J T = J ref (0), acceleration a = a(t) in terms of R T , V T , A T , r(t), and v(t).
and S T = S ref (0). The five parameters in equation (1) are referred to
as the target parameters, since they provide the path the LM The result found in Problem 9 provides the acceleration (force)
should follow. required to keep the LM in its reference trajectory.
The variable r ref in (1) represents the intended position of the LM 10. When riding in an elevator, the sensation one feels just before
at time t before the end of the landing phase. The engineers the elevator stops at a floor is jerk. Would you want jerk to be
specified the end of the landing phase to take place at t = 0, so that small or large in an elevator? Explain. Would you want jerk to
during the phase, t was always negative. Note that the LM was be small or large on a roller coaster ride? Explain. How would
landing in three dimensions, so there were actually three equations you explain snap?
like (1). Since each of those equations had this same form, we will
work in one dimension, assuming, for example, that r represents the
distance of the LM above the surface of the Moon.
A. R. Klumpp, “Apollo Lunar-Descent Guidance,” MIT Charles Stark
∗
1. If the LM follows the reference trajectory, what is the reference Draper Laboratory, R-695, June 1971,
velocity v ref (t)? http://www.hq.nasa.gov/alsj/ApolloDescentGuidnce.pdf

Chapter Review

THINGS TO KNOW

2.1 Rates of Change and the Derivative • Rate of change of a function If y = f (x), the derivative f (c)
′
is the rate of change of f with respect to x at c. (p. 173)
• Definition Derivative of a function f at a number c
• Physical If the signed distance s from the origin at time t
of an object moving on a line is given by the position
f (x) − f (c)
′
′
Form (1) f (c) = lim function s = f (t), the derivative f (t 0 ) is the velocity of the
x→c x − c object at time t 0 . (p. 173)
provided the limit exists. (p. 173)
2.2 The Derivative as a Function
Three Interpretations of the Derivative • Definition of a derivative function
• Geometric If y = f (x), the derivative f (c) f (x + h) − f (x)
′
is the slope of the tangent line to the graph of f at the Form (2) f (x) = lim
′
point (c, f (c)). (p. 173) h→0 h
provided the limit exists. (p. 179)
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