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(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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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(a) is fine, but I'm confused with (b). Please help (b)

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
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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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. =
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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