Find the particle's horizontal position x(t) and velocity v(x) at any point in a fluid whose drag force is expressed as Fdrag = kmv where, k is a constant, m is the mass of the particle and v is its velocity. Consider that the particle is initially traveling with a velocity v0.
Problem
Find the particle's horizontal position x(t) and velocity v(x) at any point in a fluid whose drag force is expressed as
Fdrag = kmv
where, k is a constant, m is the mass of the particle and v is its velocity. Consider that the particle is initially traveling with a velocity v0.
Solution:
a) To solve for the position as a function of time x(t), we construct the net force in the x-axis as
∑F = -F_____ = m_____
Then:
-m_____v = m_____
since:
a = dv/dt
then
-m_____v = m_____
by integrating, we obtain the following expression:
_____ = v0e_____
Further, employing the rules of integration results to the following expression for position as a function of time
x= (v0/_____)( _____ - e _____ )
as t -> ∞ , the position becomes
x = v0/k
b) To solve for the velocity as a function of position v(x), we construct the net force in the x-axis as follows
∑F = -F _____= m_____
Then:
-m_____v = m_____
since:
a = dv/dt
then
-m_____v = m_____
We can eliminate time by expressing, the velocity on the left side of the equation as
v = dx/dt
Then, we arrive at the following expression
_____/_____ = -k
By integrating and applying the limits, we arrive at the following
_____ = v0 - _____
which, sows that velocity decreases in a linear maner.
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