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 isits velocity. Consider that the particle is initially traveling with a velocity vo. Solution: a) To solve for the position as a function of time x(t), we construct the net force in the x-axis as E F =-F = m Then: -m y = m since: a = dv/dt then -m V = m by integrating, we obtain the following expression: s voe Further, employing the rules of integration results to the following expression for position as a function of time x= (vo/ as t+0, the position becomes x= vo/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 Σ = m Then: -m V= m
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 isits velocity. Consider that the particle is initially traveling with a velocity vo. Solution: a) To solve for the position as a function of time x(t), we construct the net force in the x-axis as E F =-F = m Then: -m y = m since: a = dv/dt then -m V = m by integrating, we obtain the following expression: s voe Further, employing the rules of integration results to the following expression for position as a function of time x= (vo/ as t+0, the position becomes x= vo/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 Σ = m Then: -m V= m
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Transcribed Image Text:x = vo/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
= m
Then:
-m
y = 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
= VO
which, sows that velocity decreases in a linear maner.
Transcribed Image Text: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 isits velocity. Consider that the particle is initially traveling with a velocity
vo.
Solution:
a) To solve for the position as a function of time x(t), we construct the net force in the x-axis as
E F =-F
= m
Then:
-m
since:
3 = dv/dt
a
then
-m
V= m
by integrating, we obtain the following expression:
= voe
Further, employing the rules of integration results to the following expression for position as a function of time
x= (vo/
as t+0, the position becomes
x = vo/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
= m
Then:
-m
V = m
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