Concept explainers
a.
Show that the function ,
a.
Explanation of Solution
Given information: If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body velocity v(t) is modelled by the initial value problem.
Differential equation:
Initial condition:
Proof:
To verify that the differential equation
Now find its derivative,
Given that
Substitute this into
Substitute
Since
The last thing to check is that the initial condition of
Hence, proved.
b.
To find the body’s limiting velocity,
b.
Answer to Problem 58E
Explanation of Solution
Given Information: : If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body velocity v(t) is modelled by the initial value problem.
Differential equation:
Initial condition:
Calculation:
As per the given problem,
Hence,
c.
To find the driver’s limiting velocity in feet per second? In miles per hours.
c.
Answer to Problem 58E
The driver’s limiting velocity is 178.9 feet per second and 122 miles per hours.
Explanation of Solution
Given Information: If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body velocity v(t) is modelled by the initial value problem.
Differential equation:
Initial condition:
For a 160-lb skydiver (mg=160), and with time in seconds and distance in feet, a typical value for k is 0.005.
Calculation:
As per the given problem,
178.9 feet per second.
122 miles per hours.
The driver’s limiting velocity is 178.9 feet per second and 122 miles per hours.
Chapter 7 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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