- #1
skeer
- 17
- 0
The Problem:
A particle moves in a medium influenced by a retarding force mk(v^3+va^2), where k and a are constants.
Show that
for any initial velocity the particle will never move more than (pi)/(2ka)
and
that it comes to rest only for t -> infinity
Attempt to solution:
I know I have to integrate the retarding force [F=-mk(v^3+va^2))] so that I can solve for v. Later, I have to solve for the constant C of the indefinite integral. Afterwards, I have to integrate to find the position.
To show the first part, I believe I have to find the value of the position as t-> infinity.
To show the second part, I believe I have to find the value of the velocity as t->infinity.
My problem: I cannot solve for v
I have solved the integral of the retarding force using trig substitution (v=acot(theta)) (dv=-acsc^2(theta)d(theta))
My final answer is:
t=(-1/ka^2)(ln(v/(sqrt(v^2+a^2))+C
If I use the quadratic equation to solve for v, I get imaginary numbers...
Am I making a mistake in my calculus or algebra?
or
Am I missing a physical concept?
A particle moves in a medium influenced by a retarding force mk(v^3+va^2), where k and a are constants.
Show that
for any initial velocity the particle will never move more than (pi)/(2ka)
and
that it comes to rest only for t -> infinity
Attempt to solution:
I know I have to integrate the retarding force [F=-mk(v^3+va^2))] so that I can solve for v. Later, I have to solve for the constant C of the indefinite integral. Afterwards, I have to integrate to find the position.
To show the first part, I believe I have to find the value of the position as t-> infinity.
To show the second part, I believe I have to find the value of the velocity as t->infinity.
My problem: I cannot solve for v
I have solved the integral of the retarding force using trig substitution (v=acot(theta)) (dv=-acsc^2(theta)d(theta))
My final answer is:
t=(-1/ka^2)(ln(v/(sqrt(v^2+a^2))+C
If I use the quadratic equation to solve for v, I get imaginary numbers...
Am I making a mistake in my calculus or algebra?
or
Am I missing a physical concept?