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Homework Statement
A block of mass ##m = 1.00 kg## is being dragged through some viscous fluid by
an external force ##F = 10.0 N##. The resistive force can be written as ##R = bv##,
where ##v## is the speed and ##b = 4.00 kg/s## is a phenomenological constant. You
may ignore gravity (we imagine that the block is floating inside the fluid). The
following integrals may come in handy:
$$\int \frac 1 { z } dz = ln(z), \int e^{at} dt = \frac 1 a e^{at}$$
a) Using the three physical quantities provided, use dimensional analysis to
find the terminal velocity of the motion ##v_T## and the characteristic time ##τ## .
b) Assuming that the block starts from rest, find ##v(t)##.
c) Find the distance traveled as a function of time ##x(t)##.
Some of the work done by the force F on the system becomes internal energy,
some becomes kinetic energy.
d) How much internal energy has been generated by the time the
block reaches half the terminal velocity, ##v(t) = v_T/2##? How big a fraction of
the total work is that?
The Attempt at a Solution
##m=M##
##F=\frac {ML} {T^2}##
##R=\frac {LM} {T^2}##
##v_T = \frac L T = m^aF^bR^c##
##M=0=a+b+c##
##L=1=b+c##
##T=1=2b2c##
This doesn't work. So i'm kinda stuck.
I also don't know what is meant by "characteristic time".
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