The motion of an object falling from rest under gravity, subject to a drag force proportional to the square of the speed, may be modelled by the initial value problem (IVP): -=g-ev², v(0)=0. Here v is the speed of the object, t is time, g is gravitational acceleration (assumed constant) and is a constant. (a) Without solving the IVP, briefly describe how you expect v to vary with t. (b) (c) (d) (e) dv dt Show that under the changes of variable x = (Eq. 4.1) becomes: dx dr ==1-x², x(0)=0. 1 A B 1-x² 1-x 1+x 1/2 (9) v, t = (eg)¹²t, the IVP given by g X = (Eq. 4.1) The IVP given by (Eq. 4.2) is of separable type. Briefly explain what this means (you do not need to solve the IVP at this stage). arks] Find A and B such that: Using your answer to (d), solve the IVP (Eq. 4.2) and show that: 1-e-2r 1+e=2r (Eq. 4.2) AS
The motion of an object falling from rest under gravity, subject to a drag force proportional to the square of the speed, may be modelled by the initial value problem (IVP): -=g-ev², v(0)=0. Here v is the speed of the object, t is time, g is gravitational acceleration (assumed constant) and is a constant. (a) Without solving the IVP, briefly describe how you expect v to vary with t. (b) (c) (d) (e) dv dt Show that under the changes of variable x = (Eq. 4.1) becomes: dx dr ==1-x², x(0)=0. 1 A B 1-x² 1-x 1+x 1/2 (9) v, t = (eg)¹²t, the IVP given by g X = (Eq. 4.1) The IVP given by (Eq. 4.2) is of separable type. Briefly explain what this means (you do not need to solve the IVP at this stage). arks] Find A and B such that: Using your answer to (d), solve the IVP (Eq. 4.2) and show that: 1-e-2r 1+e=2r (Eq. 4.2) AS
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can I get an easy explanation
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The motion of an object falling from rest under gravity, subject to a drag force
proportional to the square of the speed, may be modelled by the initial value problem
(IVP):
(b)
(c)
Here v is the speed of the object, t is time, g is gravitational acceleration (assumed
constant) and is a constant.
(a)
Without solving the IVP, briefly describe how you expect v to vary with t.
(d)
dv
dt
(e)
=g- &v², v(0)=0.
Show that under the changes of variable x =
(Eq. 4.1) becomes:
dx
dr
=1-x², x(0)=0.
X =
(Eq. 4.1)
1/2
x-A9*".
v, T = (eg)²t, the IVP given by
g
1
A
B
+
1-x² 1-x 1+x
Using your answer to (d), solve the IVP (Eq. 4.2) and show that:
1-e-2r
1+e-²r.
The IVP given by (Eq. 4.2) is of separable type. Briefly explain what this means
(you do not need to solve the IVP at this stage).
arks]
Find A and B such that:
=
(Eq. 4.2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc5a7f44-f883-4cf2-8c1e-fbb6272dba60%2Fc06cccd6-3f4e-44f7-8234-938e24f9dcf9%2Fjgrrn1c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4
The motion of an object falling from rest under gravity, subject to a drag force
proportional to the square of the speed, may be modelled by the initial value problem
(IVP):
(b)
(c)
Here v is the speed of the object, t is time, g is gravitational acceleration (assumed
constant) and is a constant.
(a)
Without solving the IVP, briefly describe how you expect v to vary with t.
(d)
dv
dt
(e)
=g- &v², v(0)=0.
Show that under the changes of variable x =
(Eq. 4.1) becomes:
dx
dr
=1-x², x(0)=0.
X =
(Eq. 4.1)
1/2
x-A9*".
v, T = (eg)²t, the IVP given by
g
1
A
B
+
1-x² 1-x 1+x
Using your answer to (d), solve the IVP (Eq. 4.2) and show that:
1-e-2r
1+e-²r.
The IVP given by (Eq. 4.2) is of separable type. Briefly explain what this means
(you do not need to solve the IVP at this stage).
arks]
Find A and B such that:
=
(Eq. 4.2)
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