Math 1320 - Lab 3
uNiD:
Name:
Due: 09/19
Today you are going to consider problems in aeronautics and use differential equation to describe them.
Answer the questions below and be sure to explain your reasoning behind your approach to these questions
with appropriate diagrams and complete sentences. In your writing, try to be complete enough so that
someone who has completed Math 1310 would understand what you are doing and why you are doing it.
1. (10 points) Hydrogen is a light and extremely powerful rocket propellant used in the space shuttles. On
a routine inspection of a liquid hydrogen tank, it is noticed that the hydrogen level is well above the
expected amount in the tank. Upon further inspection, a leak from a neighboring tank was pinpointed
as the source of the extra hydrogen in the tank. You are brought in to help prevent the tank from
overflowing. You are provided 6 volume measurements of the liquid hydrogen in the tank right after the
leak was discovered.
time (mins)
volume (m3 )
30
56094
60
62930
90
70599
120
79204
150
88857
180
99686
You are also told that an initial empirical test on these measurements suggests the hydrogen level
increase can be modeled with an exponential function with an unknown growth rate, k. Find the value
of k using this empirical assumption. What is the initial volume of the liquid hydrogen if we assume
the leak started at the same time the measurements commenced, (i.e., volume at t = 0 min)? Find the
differential equation that models the rate of change of the liquid hydrogen in the tank?
2. (10 points) A parachute is one of the basic safety devices on any aeronautic vehicle. When a person
wearing a parachute jumps off a plane, there are two forces acting on the person, the gravitational force,
FG = −mg, and the resistance force, FR = −kv. where m is the mass, g is the acceleration due to
gravity, k is a positive constant and v is the velocity. We have assumed a coordinate system such that
the ground is at y = 0. Note that FR acts against motion.
(a) Use Newton’s Second Law to write down a differential equation that relates the velocity to the net
force on the person.
(b) Solve this differential equation for the velocity as a function of time during the fall. Assume the
initial velocity, v(0) = v0
(c) The speed of the person falling under the influence of air resistance does not increase indefinitely,
instead, it approaches a finite limiting speed or terminal speed. What is this terminal speed, vτ of
the person? Draw a slope field for the differential equation you obtained in part (a).
(d) Derive the equation that describes the vertical height, y, of the person during the fall. Assume the
initial vertical position y(0) = y0
3. (10 points) According to Newton’s Law of Universal Gravitation, the gravitational force on an object of
mass m that has been projected vertically upward from the earth’s surface is
F =
mgR2
(x + R)2
where x = x(t) is the object’s distance above the surface at time t, R is the earth’s radius, and g is the
acceleration due to gravity. Also, by Newton’s Second Law,
F = ma = m(dv/dt)
and so
m
dv
mgR2
=−
dt
(x + R)2
(a) Suppose a rocket is fired vertically upward with an initial velocity v0 . Let h be the maximum height
above the surface reached by the object. Show that
r
2gRh
v0 =
R+h
[Hint: By the chain rule, m(dv/dt) = mv(dv/dx).]
(b) Calculate ve = lim v0 . This limit is called the escape velocity for the earth.
h→∞
(c) Use R = 3960 mi and g = 32 ft/s2 to calculate ve in feet per second and in miles per second.