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

Section 2.4 • Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives 215

PAGE 93. Motion on a Line As an object moves on a line,
208 91. Ideal Gas Law The Ideal Gas Law, used in chemistry and
thermodynamics, relates the pressure p, the volume V , and the its signed distance s from the origin at time t is given by the
3
absolute temperature T (in Kelvin) of a gas, using the position function s = s(t) = t − t + 1, where s is in meters and t
equation pV = nRT , where n is the amount of gas (in moles) is in seconds.
and R = 8.31 is the ideal gas constant. In an experiment, a (a) Find the velocity v, acceleration a, jerk J, and snap S of the
spherical gas container of radius r meters is placed in a pressure object at time t.
chamber and is slowly compressed while keeping its temperature
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at 273 K. (b) When is the velocity of the object 0 m/s?
(c) Find the acceleration of the object at t = 2 and at t = 5.
(a) Find the rate of change of the pressure p with respect to the 3
(d) Does the jerk of the object ever equal 0 m/s ?
radius r of the chamber.
4 (e) How would you interpret the snap for this object in rectilinear
3
Hint: The volume V of a sphere is V = πr .
3 motion?
(b) Interpret the sign of the answer found in (a).
94. Motion on a Line As an object moves on a line,
(c) If the sphere contains 1.0 mol of gas, find the rate of change
its signed distance s from the origin at time t is given by the
1
of the pressure when r = m. 1 4 2 1
4 position function s = s(t) = t − t + t + 4, where s is in
Note: The metric unit of pressure is the pascal, Pa. meters and t is in seconds. 6 2
92. Body Density The density ρ of an object is its mass m divided (a) Find the velocity v, acceleration a, jerk J, and snap S of the
m object at any time t.
by its volume V ; that is, ρ = . If a person dives below the
V (b) Find the velocity of the object at t = 0 and at t = 3.
surface of the ocean, the water pressure on the diver will steadily
(c) Find the acceleration of the object at t = 0. Interpret your
increase, compressing the diver and therefore increasing body
answer.
density. Suppose the diver is modeled as a sphere of radius r.
(d) Is the jerk of the object constant? In your own words, explain
(a) Find the rate of change of the diver’s body density with what the jerk says about the acceleration of the object.
respect to the radius r of the sphere. (e) How would you interpret the snap for this object in rectilinear
4 motion?
3
Hint: The volume V of a sphere is V = πr .
3
95. Elevator Ride Quality The ride quality of an elevator depends
(b) Interpret the sign of the answer found in (a).
on several factors, two of which are acceleration and jerk. In a
(c) Find the rate of change of the diver’s body density when the study of 367 persons riding in a 1600-kg elevator that moves at an
radius is 45 cm and the mass is 80,000 g (80 kg). average speed of 4 m/s, the majority of riders were comfortable in
an elevator with vertical motion given by
Jerk and Snap Problems 93–96 use the following discussion:
Suppose that an object is moving on a line so that its signed 2 3
s(t) = 4t + 0.8t + 0.333t
distance s from the origin at time t is given by the position
function s = s(t). The velocity v = v(t) of the object at time t
(a) Find the acceleration that the riders found acceptable.
is the rate of change of s with respect to time,
ds (b) Find the jerk that the riders found acceptable.
namely, v = v(t) = . The acceleration a = a(t) of the object at
dt Source: Elevator Ride Quality, January 2007, http://www
time t is the rate of change of the velocity with respect to time, .lift-report.de/index.php/news/176/368/Elevator-Ride-Quality
2
dv d ds d s 96. Elevator Ride Quality In a hospital, the effects of high
a = a(t) = = = 2
dt dt dt dt acceleration or jerk may be harmful to patients, so the
acceleration and jerk need to be lower than in standard elevators.
There are also physical interpretations of the third derivative It has been determined that a 1600-kg elevator that is installed in
and the fourth derivative of s = s(t). The jerk J = J(t) of the a hospital and that moves at an average speed of 4 m/s should
object at time t is the rate of change of the acceleration a with have vertical motion
respect to time; that is,
2 3
s(t) = 4t + 0.55t + 0.1167t
2 3
da d dv d v d s
J = J(t) = = = =
dt dt dt dt 2 dt 3 (a) Find the acceleration of a hospital elevator.
The snap S = S(t) of the object at time t is the rate of change of (b) Find the jerk of a hospital elevator.
the jerk J with respect to time; that is,
Source: Elevator Ride Quality, January 2007, http://www.lift
2
3
4
d J d a d v d s -report.de/index.php/news/176/368/Elevator-Ride-Quality
S = S(t) = = = =
dt dt 2 dt 3 dt 4
Engineers take jerk into consideration when designing
elevators, aircraft, and cars. In these cases, they try to minimize
jerk, making for a smooth ride. But when designing thrill rides,
such as roller coasters, the jerk is increased, making for an
exciting experience.
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