In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is mdv = mg - kv, dt where k> 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0) = Vo -kt v(t) = gm k +V V- gm k (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. gm k gm s(t) = 1 + k (c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0. -) (1 -e ( =))) m k V 0 X - gm k X
In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is mdv = mg - kv, dt where k> 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0) = Vo -kt v(t) = gm k +V V- gm k (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. gm k gm s(t) = 1 + k (c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0. -) (1 -e ( =))) m k V 0 X - gm k X
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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v(t) = g
In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is
dv
dt
m-
= mg - kv,
where k > 0 is a constant of proportionality. The positive direction is downward.
(a) Solve the equation subject to the initial condition v(0) = Vo
+V
k
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s(t) = gm
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0
k
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gm
k
(b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass.
gm
k
-kt
m
ZILLDIFFEQMODAP11 3.1.035.
X
(c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0.
30-²+(-²)(1-))
gm
(⇒))
k
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X
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Step 1: Write general form of linear differential equation
VIEWStep 2: Solving the velocity w.r.t the initial condition
VIEWStep 3: Solving the velocity w.r.t the initial condition
VIEWStep 4: Finding the terminal velocity
VIEWStep 5: Finding the distance s(t) w.r.t. the initial condition
VIEWStep 6: Finding the distance s(t) w.r.t. the initial condition
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