Solved: An object moving in a liquid experiences a linear

QUESTION:

An object moving in a liquid experiences a linear drag force: \(\vec{D}=(b v\), direction opposite the motion ), where \(b\) is a constant called the drag coefficient. For a sphere of radius \(R\), the drag constant can be computed as \(b=6 \pi \eta R\), where \(\eta\) is the viscosity of the liquid.

a. Use what you've learned in calculus to prove that

                                                           \(a_{x}=v_{x} \frac{d v_{x}}{d x}\)

b. Find an algebraic expression for \(v_{x}(x)\), the \(x\)-component of velocity as a function of distance traveled, for a spherical particle of radius \(R\) and mass \(m\) that is shot horizontally with initial speed \(v_{0}\) through a liquid of viscosity \(\eta\).

c. Water at \(20^{\circ} \mathrm{C}\) has viscosity \(\eta=1.0 \times 10^{-3} \mathrm{Ns} / \mathrm{m}^{2}\). Suppose a 1.0-cm-diameter,1.0 g marble is shot horizontally into a tank of \(20^{\circ} \mathrm{C}\) water at \(10 \mathrm{~cm} / \mathrm{s}\). How far will it travel before stopping?

Equation Transcription:

,direction opposite  the motion)

Text Transcription:

^vec D=(bv,direction opposite  the motion)

b

R

b=6 pi eta R

eta

a_x=v_x dv_x/dx

v_x(x)

x-components

R

m

v_0

eta

20^circ C

eta=1.0 x 10^-3 N s/m^2

20^circ C