(b) Suppose the raindrop accretes mass at rate dm/dt = bmv with b > 0. i) Show that the speed of the raindrop falling under the force of gravity at time t is v (t) = √ √ 7/1 tanh(V/gbt), with initial condition at t = 0, v(0) = 0. ii) Compute m(t) given the initial conditions that at t = Hint: The following integral may be of use So tanh(at')dt' = 0, m(0) = mį. ¹ In(cosh(at)). a
(b) Suppose the raindrop accretes mass at rate dm/dt = bmv with b > 0. i) Show that the speed of the raindrop falling under the force of gravity at time t is v (t) = √ √ 7/1 tanh(V/gbt), with initial condition at t = 0, v(0) = 0. ii) Compute m(t) given the initial conditions that at t = Hint: The following integral may be of use So tanh(at')dt' = 0, m(0) = mį. ¹ In(cosh(at)). a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) is fine, but I'm confused with (b). Please help (b)
Transcribed Image Text:A raindrop falling through clouds is accumulating mass.
(a) Explain the origin of the equation of motion for mass accretion
dv
m + v
dt
and define all the quantities in the above equation.
=
(b) Suppose the raindrop accretes mass at rate dm/dt - bmv with b > 0.
i) Show that the speed of the raindrop falling under the force of gravity at time t is
•t
S
dm
dt
=
F,
with initial condition at t = 0, v(0) = 0.
ii) Compute m(t) given the initial conditions that at t = 0, m(0) = mį.
Hint: The following integral may be of use
v(t) = √² tanh(V/gbt),
b
1
tanh(at')dt' = ¹ In(cosh(at)).
a
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Follow-up Question
Transcribed Image Text:Suppose the raindrop falling through the clouds accretes mass at rate dm(t)/dt
yr(t)², where r(t) is the radius of the raindrop at time t with r(0) = a, y > 0, and
m(t) = 4ñr³(t)p/3 with p the density of water.
i) Show that r(t)
=
At + C and determine A and C.
=
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