ODE problem Your physics class project is to measure the temperature in the atmo

Question

ODE problem
Your physics class project is to measure the temperature in the atmosphere at dierent
heights above the earth's surface using a weather balloon that contains an apparatus that
can record and store temperature over a period of time. You were all ready to let the
balloon
y, and you wondered where the balloon would land, so you can collect the tem-
perature data. Unfortunately, it was a gloomy day and you did not have any hope of seeing
where the balloon
ew. In order to estimate the landing location, you set up two wind
speed meters (anemometers). One measures the wind speed in the West-East direction.
A positive value for the wind means that is is blowing from the West towards the East
(positive x-direction). The other anemometer measures the wind speed in the South-North
direction, with positive values meaning the wind is blowing from the South towards the
North (positive y-direction).
The measurements are shown in Figure 1. You were able to t the wind speed data
with functions that match the measurements fairly closely.
0 1000 2000 3000 4000 5000 6000 7000 8000
?0.5
0
0.5
1
1.5
2
Time t [s]
Wind Speed [m/s]
Anemometer West?East
0 1000 2000 3000 4000 5000 6000 7000 8000
?6
?4
?2
0
2
4
Time t [s]
Wind Speed [m/s]
Anemometer South?North
Figure 1: Measured speeds of wind (blue) and t (red) with formulas in Equations (1) and
(2).
.
1
The formula you used for tting the wind speed in the West-East direction is
vx;ground = 2 atan(
t
7200
); (1)
and the formula you used for tting the wind speed in the South-North direction is
vy;ground = 5 sin3(
t
600
) sin(
t
7200
): (2)
The measurements were taken at the location where you let go of the balloon. You
know that the winds in the South-North direction blow stronger the further East you go.
Therefore you adjust the formula for the wind speed in the South-North direction to
vy;ground = 5 sin3(
t
600
) sin(
t
7200
)
| {z }
at site of takeo
(1 +
3
1000
x)
| {z }
correct for East-West location
: (3)
Additionally, you know that the winds will blow stronger the higher up in the atmo-
sphere that you go. If you denote the vertical position of the balloon with the z-coordinate,
then you can adjust the equations for the wind speed in the x- and y-directions to:
vx = 2 atan(
t
7200
)
| {z }
on ground
(1 +
5
1000
z)
| {z }
correct for altitude
;
vy = 5 sin3(
t
600
) sin(
t
7200
) (1 +
3
1000
x)
| {z }
on ground, variable in East-West
(1 +
5
1000
z)
| {z }
correct for altitude
:
(4)
(5)
In order to estimate the vertical position of the balloon you assume that the vertical
(upward) velocity vz of the balloon is given by the following formula:
vz = 1:2 cos3(
t
7200
) (6)
You need to calculate the trajectory of the balloon to get an estimate of where it might
have landed. If the position of the balloon is stored in the vector P = (x; y; z), then the
velocity vector v = (vx; vy; vz) of the balloon is simply the derivative of the position with
respect to time:
d
dt
P = v(t) (7)
or in another (maybe more familiar) format:
dx
dt = vx(t)
dy
dt = vy(t)
dz
dt = vz(t)
(8)
2
This is a system of rst order ODEs for the position P = (x; y; z) of the balloon. You
will now solve this ODE with Matlab. Note here that v is a function of time and the
current position P of the balloon.
Write a function balloonODE(t,P) that takes time t and position P as inputs and
returns the velocities vx,vy and vz in a vector v = (vx; vy; vz). Use Equations (4),(5) and
(6) for this . Store the function in a le called balloonODE.m.
Simulate the trajectory of your balloon. For this, write a script in a le called
balloonFlight.m. In this script, set the initial position of the balloon to
P0 =
2
4
0
0
0
3
5: (9)
Solve the ODE for the time interval from 0 to 7200 seconds by calling ode23 in Matlab.
Make sure to store the solution of the position of the balloon in the variable P. Then display
the
ight of the balloon with the plot commands
X = P(:,1);
Y = P(:,2);
Z = P(:,3);
figure
comet3(X,Y,Z)
plot3(X,Y,Z)

Explanation / Answer

This seems hard indeed. First thing that needs to be done is identify the question(s) Solve the ODE for the time interval from 0 to 7200 seconds by calling ode23 in Matlab. Remember a ordinary differential equation (ODE) is --- "is an equation in which there is only one independent variable and one or more derivatives of a dependent variable with respect to the independent variable, so that all the derivatives occurring in the equation are ordinary derivatives." thanks to Wikipedia I really hope this helped even if just a little, best of luck.

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