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 

F_drag = 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.

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 

F_drag = 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.

To solve for the position as a function of time x(t), construct the net force in the x-axis.

Further, employ the rules of integration results to the following expression for position as a function of time.

To solve for the velocity as a function of position v(x), we construct the net force in the x-axis, then, eliminate time by expressing, the velocity on the left side of the equation.

By integrating and applying the limits, we then shall arrive with a velocity that decreases in a linear maner.