Motion Under a Force that is a Function of Velocity. (a) A fluid flows with a un
ID: 1654713 • Letter: M
Question
Motion Under a Force that is a Function of Velocity. (a) A fluid flows with a uniform speed v_f. A particle of mass m is placed within the fluid and given an initial speed v_0. Given that the magnitude of the force on the particle at time t is proportional to [v_f - v(t)], determine the speed of the particle as a function of time. Consider all possible cases you can think of. Describe the motion of the particle (i.e. use words!). b) The speed v_f (the speed of the fluid) has another physical meaning: what is this second meaning of v_f?Explanation / Answer
a. fluid flow velocity = vf
mass of particel = m
partticle velocity = v(t)
then Force on particle = k(vf - v(t)), where k is a constant
so from newtons second law
mdv(t)/dt = k(vf - v(t))
dv(t)/(vf - v(t)) = k*dt/m
integrating v(t) from vo to v
ln([vf - v]/[vf - vo]) = kt/m
vf - v = [vf - vo]e^(kt/m)
v = vf - [vf - vo]e^(kt/m)
b. vf, the flow velocity is nothing but a parameter to make a physically practical situation of a particle moviong in moving medium. this also means that even if the m is not moving, the relative motion of the particel and the medium is more important than the absolute moption of the medium
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