You are given that a very simple mathematical model of the motion of a sailboat in wavy waters is the following ordinary differential equation 100 Q 1 d²x dt² = 50Vwind Cos (0) + 6 Awaves sin X 10 dx 60- " dt is the angle in which the wind is blowing, and Awaves is the = where Vwind is the wind speed, amplitude of the waves. When 0 0° the wind is blowing in the positive x direction (i.e. direction of travel of the sailboat) and when 0 = 180° the wind is blowing directly opposite to the positive x direction. Note that x/10 should be in radians when computing the sin() term on the right hand side of the equation. Do not worry about reporting units for the other variables (e.g. Vwind, Awaves, x, t) as they have been given in dimensionless form. Write a computer program that uses the 2nd order Runge Kutta method to solve the equation in Q3. Use this computer program to solve the equation for Vwind = 10,0 = 30° and Awaves 2. Assume that dx/dt (0) = x(0) = 0. Plot dx/dt for t = [0, 50]. Use a small value of time step size At so that you are reasonably sure that your results are accurate. Justify the value of At that you use in order to get an accurate solution. -
You are given that a very simple mathematical model of the motion of a sailboat in wavy waters is the following ordinary differential equation 100 Q 1 d²x dt² = 50Vwind Cos (0) + 6 Awaves sin X 10 dx 60- " dt is the angle in which the wind is blowing, and Awaves is the = where Vwind is the wind speed, amplitude of the waves. When 0 0° the wind is blowing in the positive x direction (i.e. direction of travel of the sailboat) and when 0 = 180° the wind is blowing directly opposite to the positive x direction. Note that x/10 should be in radians when computing the sin() term on the right hand side of the equation. Do not worry about reporting units for the other variables (e.g. Vwind, Awaves, x, t) as they have been given in dimensionless form. Write a computer program that uses the 2nd order Runge Kutta method to solve the equation in Q3. Use this computer program to solve the equation for Vwind = 10,0 = 30° and Awaves 2. Assume that dx/dt (0) = x(0) = 0. Plot dx/dt for t = [0, 50]. Use a small value of time step size At so that you are reasonably sure that your results are accurate. Justify the value of At that you use in order to get an accurate solution. -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![You are given that a very simple mathematical model of the motion of a sailboat in wavy waters is
the following ordinary differential equation
d²x
dt²
Q 1
100-
=
50Vwind Cos (0) + 6 Awaves sin
X
10.
60
=
where Vwind is the wind speed, is the angle in which the wind is blowing, and Awaves is the
amplitude of the waves. When 0 = 0º the wind is blowing in the positive à direction (i.e. direction
of travel of the sailboat) and when 0 = 180° the wind is blowing directly opposite to the positive x
direction. Note that x/10 should be in radians when computing the sin ( term on the right
hand side of the equation. Do not worry about reporting units for the other variables (e.g.
Vwind, Awaves, x, t) as they have been given in dimensionless form.
dx
dt'
=
Write a computer program that uses the 2nd order Runge Kutta method to solve the equation in
Q3. Use this computer program to solve the equation for Vwind 10, 0 = 30° and Awaves 2.
Assume that dx/dt(0) = x(0) = 0. Plot dx/dt for t = [0, 50]. Use a small value of time step size
At so that you are reasonably sure that your results are accurate. Justify the value of At that you
use in order to get an accurate solution.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12fdac56-4b52-4434-8b00-5772085cd8ad%2F3ec2c20b-a899-4c9c-a43d-5b34683d1591%2F8kaut5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:You are given that a very simple mathematical model of the motion of a sailboat in wavy waters is
the following ordinary differential equation
d²x
dt²
Q 1
100-
=
50Vwind Cos (0) + 6 Awaves sin
X
10.
60
=
where Vwind is the wind speed, is the angle in which the wind is blowing, and Awaves is the
amplitude of the waves. When 0 = 0º the wind is blowing in the positive à direction (i.e. direction
of travel of the sailboat) and when 0 = 180° the wind is blowing directly opposite to the positive x
direction. Note that x/10 should be in radians when computing the sin ( term on the right
hand side of the equation. Do not worry about reporting units for the other variables (e.g.
Vwind, Awaves, x, t) as they have been given in dimensionless form.
dx
dt'
=
Write a computer program that uses the 2nd order Runge Kutta method to solve the equation in
Q3. Use this computer program to solve the equation for Vwind 10, 0 = 30° and Awaves 2.
Assume that dx/dt(0) = x(0) = 0. Plot dx/dt for t = [0, 50]. Use a small value of time step size
At so that you are reasonably sure that your results are accurate. Justify the value of At that you
use in order to get an accurate solution.
=
Transcribed Image Text:Q 2
Compute the solution for Awaves
0.1, 1 and 6. Keep Vwind
10, 0 = 30°. Plot dx/dt for 0 ≤
t≤ 50 for Awaves = 0.1, 1 and 6. According to this model, what happens to the velocity, dx/dt, of
the sailboat when Awaves increases? Use the value of At that you have justified in Q3.1.
Q3.
-
=
10, Awaves 6. Use a
Compute the solution for 0 = 0º, 10°, 30°, 50° and 90°. Keep Vwind
small value of At so that you are reasonably sure that your results are accurate. You should justify
this value of At. Plot dx/dt for 0 ≤ t ≤ 50 for all values of 0. According to this model, what
happens to the velocity of the sailboat when increases? Use the value of At that you have
justified in Q3.1.
=
=
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