Fox 8ed - MF ex 1.7 PDF

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Problem 1.7

Given:

[Difficulty: 3]

Small particle accelerating from rest in a fluid. Net weight is W, resisting force FD = kV, where V is speed.

Find:

Time required to reach 95 percent of terminal speed, Vt.

Solution:

Consider the particle to be a system. Apply Newton's second law.

Basic equation: ∑Fy = may

FD = kV

Assumptions: 1.

W is net weight

2.

Resisting force acts opposite to V

Particle y

Then

W

∑F

y

= W − kV = ma y = m

dV W dV = g dt dt

or

k dV = g(1 − V) W dt

Separating variables,

dV = g dt 1 − Wk V

Integrating, noting that velocity is zero initially,

∫

or

1−

V

But V→Vt as t→∞, so Vt =

When

V Vt

= 0.95 , then e

−

kgt W

W k

. Therefore

= 0.05 and

kgt W

0

t dV W = − ln(1 − Wk V ) = ∫ gdt = gt k 0 1−WV k

kgt kgt − − ⎞ k W⎛ V = e W ; V = ⎜⎜ 1− e W ⎟⎟ W k⎝ ⎠

− V = 1− e Vt

kgt W

= 3. Thus t = 3W/gk...