C3m - Model and analyze the vertical motion of an object with linear drag

Example 1

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 1)

A water droplet with a radius of \(0.000312\) meters has a mass of about \(9.59 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.845\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.6\).

The velocity after \(0.03\) seconds is approximately \(-0.249\) meters per second.

Example 2

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 2)

A water droplet with a radius of \(0.000416\) meters has a mass of about \(2.26 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.975\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.1\).

The velocity after \(0.03\) seconds is approximately \(-0.254\) meters per second.

Example 3

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 3)

A water droplet with a radius of \(0.000104\) meters has a mass of about \(3.51 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.487\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 20.1\).

The velocity after \(0.04\) seconds is approximately \(-0.269\) meters per second.

Example 4

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 4)

A water droplet with a radius of \(0.000183\) meters has a mass of about \(1.93 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.647\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 15.2\).

The velocity after \(0.02\) seconds is approximately \(-0.169\) meters per second.

Example 5

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 5)

A water droplet with a radius of \(5.59 \times 10^{-6}\) meters has a mass of about \(5.48 \times 10^{-14}\) kilograms and a downward terminal velocity of approximately \(0.113\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 86.8\).

The velocity after \(0.03\) seconds is approximately \(-0.105\) meters per second.

Example 6

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 6)

A water droplet with a radius of \(0.000354\) meters has a mass of about \(1.39 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.899\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.9\).

The velocity after \(0.04\) seconds is approximately \(-0.318\) meters per second.

Example 7

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 7)

A water droplet with a radius of \(0.000307\) meters has a mass of about \(9.12 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.838\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.7\).

The velocity after \(0.04\) seconds is approximately \(-0.313\) meters per second.

Example 8

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 8)

A water droplet with a radius of \(0.000342\) meters has a mass of about \(1.26 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.884\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.1\).

The velocity after \(0.04\) seconds is approximately \(-0.317\) meters per second.

Example 9

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 9)

A water droplet with a radius of \(0.000180\) meters has a mass of about \(1.84 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.642\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 15.3\).

The velocity after \(0.02\) seconds is approximately \(-0.169\) meters per second.

Example 10

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 10)

A water droplet with a radius of \(0.000407\) meters has a mass of about \(2.11 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.964\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.2\).

The velocity after \(0.03\) seconds is approximately \(-0.254\) meters per second.

Example 11

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 11)

A water droplet with a radius of \(0.000127\) meters has a mass of about \(6.39 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.538\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 18.2\).

The velocity after \(0.03\) seconds is approximately \(-0.227\) meters per second.

Example 12

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 12)

A water droplet with a radius of \(0.000127\) meters has a mass of about \(6.46 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.539\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 18.2\).

The velocity after \(0.03\) seconds is approximately \(-0.227\) meters per second.

Example 13

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 13)

A water droplet with a radius of \(0.0000421\) meters has a mass of about \(2.34 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.310\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 31.6\).

The velocity after \(0.04\) seconds is approximately \(-0.223\) meters per second.

Example 14

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 14)

A water droplet with a radius of \(7.51 \times 10^{-6}\) meters has a mass of about \(1.33 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.131\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 74.9\).

The velocity after \(0.02\) seconds is approximately \(-0.102\) meters per second.

Example 15

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 15)

A water droplet with a radius of \(0.000218\) meters has a mass of about \(3.26 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.706\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.9\).

The velocity after \(0.04\) seconds is approximately \(-0.301\) meters per second.

Example 16

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 16)

A water droplet with a radius of \(0.0000987\) meters has a mass of about \(3.02 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.475\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 20.7\).

The velocity after \(0.03\) seconds is approximately \(-0.219\) meters per second.

Example 17

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 17)

A water droplet with a radius of \(0.000112\) meters has a mass of about \(4.42 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.506\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 19.4\).

The velocity after \(0.03\) seconds is approximately \(-0.223\) meters per second.

Example 18

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 18)

A water droplet with a radius of \(0.000410\) meters has a mass of about \(2.17 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.968\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.1\).

The velocity after \(0.03\) seconds is approximately \(-0.254\) meters per second.

Example 19

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 19)

A water droplet with a radius of \(0.0000871\) meters has a mass of about \(2.07 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.446\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 22.0\).

The velocity after \(0.04\) seconds is approximately \(-0.261\) meters per second.

Example 20

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 20)

A water droplet with a radius of \(0.0000979\) meters has a mass of about \(2.95 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.473\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 20.7\).

The velocity after \(0.03\) seconds is approximately \(-0.219\) meters per second.

Example 21

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 21)

A water droplet with a radius of \(0.0000454\) meters has a mass of about \(2.94 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.322\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 30.5\).

The velocity after \(0.02\) seconds is approximately \(-0.147\) meters per second.

Example 22

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 22)

A water droplet with a radius of \(0.0000471\) meters has a mass of about \(3.28 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.328\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.9\).

The velocity after \(0.03\) seconds is approximately \(-0.194\) meters per second.

Example 23

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 23)

A water droplet with a radius of \(0.0000158\) meters has a mass of about \(1.24 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.190\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 51.6\).

The velocity after \(0.02\) seconds is approximately \(-0.122\) meters per second.

Example 24

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 24)

A water droplet with a radius of \(0.0000971\) meters has a mass of about \(2.87 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.471\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 20.8\).

The velocity after \(0.04\) seconds is approximately \(-0.266\) meters per second.

Example 25

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 25)

A water droplet with a radius of \(0.000126\) meters has a mass of about \(6.24 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.536\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 18.3\).

The velocity after \(0.04\) seconds is approximately \(-0.278\) meters per second.

Example 26

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 26)

A water droplet with a radius of \(0.000364\) meters has a mass of about \(1.52 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.912\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.8\).

The velocity after \(0.04\) seconds is approximately \(-0.319\) meters per second.

Example 27

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 27)

A water droplet with a radius of \(0.0000254\) meters has a mass of about \(5.16 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.241\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 40.7\).

The velocity after \(0.04\) seconds is approximately \(-0.194\) meters per second.

Example 28

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 28)

A water droplet with a radius of \(0.000106\) meters has a mass of about \(3.73 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.492\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 19.9\).

The velocity after \(0.04\) seconds is approximately \(-0.270\) meters per second.

Example 29

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 29)

A water droplet with a radius of \(0.000304\) meters has a mass of about \(8.80 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.833\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.8\).

The velocity after \(0.03\) seconds is approximately \(-0.248\) meters per second.

Example 30

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 30)

A water droplet with a radius of \(0.0000787\) meters has a mass of about \(1.53 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.424\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 23.1\).

The velocity after \(0.04\) seconds is approximately \(-0.256\) meters per second.

Example 31

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 31)

A water droplet with a radius of \(4.83 \times 10^{-6}\) meters has a mass of about \(3.53 \times 10^{-14}\) kilograms and a downward terminal velocity of approximately \(0.105\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 93.4\).

The velocity after \(0.03\) seconds is approximately \(-0.0986\) meters per second.

Example 32

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 32)

A water droplet with a radius of \(0.000141\) meters has a mass of about \(8.84 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.568\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 17.3\).

The velocity after \(0.03\) seconds is approximately \(-0.230\) meters per second.

Example 33

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 33)

A water droplet with a radius of \(8.46 \times 10^{-6}\) meters has a mass of about \(1.90 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.139\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 70.6\).

The velocity after \(0.03\) seconds is approximately \(-0.122\) meters per second.

Example 34

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 34)

A water droplet with a radius of \(0.000189\) meters has a mass of about \(2.12 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.657\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 14.9\).

The velocity after \(0.04\) seconds is approximately \(-0.295\) meters per second.

Example 35

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 35)

A water droplet with a radius of \(0.0000479\) meters has a mass of about \(3.46 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.331\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.6\).

The velocity after \(0.04\) seconds is approximately \(-0.230\) meters per second.

Example 36

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 36)

A water droplet with a radius of \(0.000324\) meters has a mass of about \(1.07 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.861\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.4\).

The velocity after \(0.03\) seconds is approximately \(-0.249\) meters per second.

Example 37

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 37)

A water droplet with a radius of \(0.000275\) meters has a mass of about \(6.50 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.792\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.4\).

The velocity after \(0.04\) seconds is approximately \(-0.309\) meters per second.

Example 38

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 38)

A water droplet with a radius of \(0.0000381\) meters has a mass of about \(1.74 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.295\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 33.3\).

The velocity after \(0.02\) seconds is approximately \(-0.143\) meters per second.

Example 39

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 39)

A water droplet with a radius of \(0.000294\) meters has a mass of about \(7.95 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.819\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.0\).

The velocity after \(0.02\) seconds is approximately \(-0.174\) meters per second.

Example 40

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 40)

A water droplet with a radius of \(0.0000813\) meters has a mass of about \(1.69 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.431\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 22.8\).

The velocity after \(0.03\) seconds is approximately \(-0.213\) meters per second.

Example 41

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 41)

A water droplet with a radius of \(0.0000666\) meters has a mass of about \(9.27 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.390\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 25.2\).

The velocity after \(0.04\) seconds is approximately \(-0.247\) meters per second.

Example 42

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 42)

A water droplet with a radius of \(0.0000809\) meters has a mass of about \(1.66 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.430\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 22.8\).

The velocity after \(0.04\) seconds is approximately \(-0.257\) meters per second.

Example 43

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 43)

A water droplet with a radius of \(0.0000386\) meters has a mass of about \(1.81 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.297\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 33.0\).

The velocity after \(0.04\) seconds is approximately \(-0.218\) meters per second.

Example 44

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 44)

A water droplet with a radius of \(8.70 \times 10^{-6}\) meters has a mass of about \(2.07 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.141\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 69.6\).

The velocity after \(0.02\) seconds is approximately \(-0.106\) meters per second.

Example 45

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 45)

A water droplet with a radius of \(0.000316\) meters has a mass of about \(9.93 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.850\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.5\).

The velocity after \(0.03\) seconds is approximately \(-0.249\) meters per second.

Example 46

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 46)

A water droplet with a radius of \(0.0000743\) meters has a mass of about \(1.29 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.412\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 23.8\).

The velocity after \(0.04\) seconds is approximately \(-0.253\) meters per second.

Example 47

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 47)

A water droplet with a radius of \(0.000325\) meters has a mass of about \(1.08 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.862\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.4\).

The velocity after \(0.03\) seconds is approximately \(-0.249\) meters per second.

Example 48

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 48)

A water droplet with a radius of \(0.0000267\) meters has a mass of about \(5.98 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.247\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 39.7\).

The velocity after \(0.03\) seconds is approximately \(-0.172\) meters per second.

Example 49

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 49)

A water droplet with a radius of \(0.000129\) meters has a mass of about \(6.68 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.542\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 18.1\).

The velocity after \(0.02\) seconds is approximately \(-0.165\) meters per second.

Example 50

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 50)

A water droplet with a radius of \(0.000160\) meters has a mass of about \(1.28 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.604\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 16.2\).

The velocity after \(0.02\) seconds is approximately \(-0.168\) meters per second.

Example 51

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 51)

A water droplet with a radius of \(0.0000482\) meters has a mass of about \(3.53 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.332\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.5\).

The velocity after \(0.02\) seconds is approximately \(-0.148\) meters per second.

Example 52

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 52)

A water droplet with a radius of \(0.0000721\) meters has a mass of about \(1.18 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.406\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 24.2\).

The velocity after \(0.04\) seconds is approximately \(-0.252\) meters per second.

Example 53

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 53)

A water droplet with a radius of \(0.000253\) meters has a mass of about \(5.07 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.760\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.9\).

The velocity after \(0.03\) seconds is approximately \(-0.244\) meters per second.

Example 54

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 54)

A water droplet with a radius of \(0.000224\) meters has a mass of about \(3.52 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.715\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.7\).

The velocity after \(0.03\) seconds is approximately \(-0.241\) meters per second.

Example 55

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 55)

A water droplet with a radius of \(0.000417\) meters has a mass of about \(2.28 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.976\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.1\).

The velocity after \(0.04\) seconds is approximately \(-0.323\) meters per second.

Example 56

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 56)

A water droplet with a radius of \(0.000175\) meters has a mass of about \(1.69 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.633\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 15.5\).

The velocity after \(0.02\) seconds is approximately \(-0.169\) meters per second.

Example 57

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 57)

A water droplet with a radius of \(0.000236\) meters has a mass of about \(4.12 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.734\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.4\).

The velocity after \(0.02\) seconds is approximately \(-0.172\) meters per second.

Example 58

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 58)

A water droplet with a radius of \(0.000211\) meters has a mass of about \(2.94 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.694\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 14.1\).

The velocity after \(0.03\) seconds is approximately \(-0.240\) meters per second.

Example 59

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 59)

A water droplet with a radius of \(0.0000351\) meters has a mass of about \(1.35 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.283\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 34.7\).

The velocity after \(0.02\) seconds is approximately \(-0.142\) meters per second.

Example 60

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 60)

A water droplet with a radius of \(0.0000471\) meters has a mass of about \(3.28 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.328\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.9\).

The velocity after \(0.03\) seconds is approximately \(-0.194\) meters per second.

Example 61

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 61)

A water droplet with a radius of \(0.0000161\) meters has a mass of about \(1.32 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.192\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 51.1\).

The velocity after \(0.04\) seconds is approximately \(-0.167\) meters per second.

Example 62

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 62)

A water droplet with a radius of \(9.20 \times 10^{-6}\) meters has a mass of about \(2.45 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.145\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 67.7\).

The velocity after \(0.03\) seconds is approximately \(-0.126\) meters per second.

Example 63

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 63)

A water droplet with a radius of \(0.000317\) meters has a mass of about \(1.00 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.851\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.5\).

The velocity after \(0.02\) seconds is approximately \(-0.175\) meters per second.

Example 64

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 64)

A water droplet with a radius of \(0.000393\) meters has a mass of about \(1.91 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.948\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.3\).

The velocity after \(0.04\) seconds is approximately \(-0.321\) meters per second.

Example 65

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 65)

A water droplet with a radius of \(0.000278\) meters has a mass of about \(6.75 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.797\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.3\).

The velocity after \(0.02\) seconds is approximately \(-0.174\) meters per second.

Example 66

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 66)

A water droplet with a radius of \(0.000110\) meters has a mass of about \(4.21 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.502\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 19.5\).

The velocity after \(0.02\) seconds is approximately \(-0.162\) meters per second.

Example 67

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 67)

A water droplet with a radius of \(0.000389\) meters has a mass of about \(1.85 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.943\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.4\).

The velocity after \(0.04\) seconds is approximately \(-0.321\) meters per second.

Example 68

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 68)

A water droplet with a radius of \(0.0000256\) meters has a mass of about \(5.29 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.242\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 40.5\).

The velocity after \(0.04\) seconds is approximately \(-0.194\) meters per second.

Example 69

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 69)

A water droplet with a radius of \(0.000110\) meters has a mass of about \(4.21 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.502\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 19.5\).

The velocity after \(0.03\) seconds is approximately \(-0.223\) meters per second.

Example 70

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 70)

A water droplet with a radius of \(0.000366\) meters has a mass of about \(1.55 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.915\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.7\).

The velocity after \(0.03\) seconds is approximately \(-0.252\) meters per second.

Example 71

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 71)

A water droplet with a radius of \(0.000296\) meters has a mass of about \(8.18 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.823\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.9\).

The velocity after \(0.03\) seconds is approximately \(-0.247\) meters per second.

Example 72

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 72)

A water droplet with a radius of \(0.000288\) meters has a mass of about \(7.49 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.811\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.1\).

The velocity after \(0.04\) seconds is approximately \(-0.311\) meters per second.

Example 73

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 73)

A water droplet with a radius of \(0.000226\) meters has a mass of about \(3.64 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.719\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.6\).

The velocity after \(0.03\) seconds is approximately \(-0.242\) meters per second.

Example 74

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 74)

A water droplet with a radius of \(0.0000423\) meters has a mass of about \(2.38 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.311\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 31.5\).

The velocity after \(0.03\) seconds is approximately \(-0.190\) meters per second.

Example 75

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 75)

A water droplet with a radius of \(0.0000750\) meters has a mass of about \(1.33 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.414\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 23.7\).

The velocity after \(0.03\) seconds is approximately \(-0.211\) meters per second.

Example 76

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 76)

A water droplet with a radius of \(0.000370\) meters has a mass of about \(1.59 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.919\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.7\).

The velocity after \(0.03\) seconds is approximately \(-0.252\) meters per second.

Example 77

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 77)

A water droplet with a radius of \(0.000383\) meters has a mass of about \(1.77 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.936\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 10.5\).

The velocity after \(0.04\) seconds is approximately \(-0.321\) meters per second.

Example 78

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 78)

A water droplet with a radius of \(0.0000397\) meters has a mass of about \(1.96 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.301\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 32.6\).

The velocity after \(0.04\) seconds is approximately \(-0.219\) meters per second.

Example 79

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 79)

A water droplet with a radius of \(0.0000589\) meters has a mass of about \(6.43 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.367\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 26.7\).

The velocity after \(0.02\) seconds is approximately \(-0.152\) meters per second.

Example 80

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 80)

A water droplet with a radius of \(0.000281\) meters has a mass of about \(7.01 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.802\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.2\).

The velocity after \(0.04\) seconds is approximately \(-0.310\) meters per second.

Example 81

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 81)

A water droplet with a radius of \(0.000348\) meters has a mass of about \(1.33 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.892\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.0\).

The velocity after \(0.02\) seconds is approximately \(-0.176\) meters per second.

Example 82

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 82)

A water droplet with a radius of \(0.000302\) meters has a mass of about \(8.67 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.831\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.8\).

The velocity after \(0.02\) seconds is approximately \(-0.175\) meters per second.

Example 83

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 83)

A water droplet with a radius of \(0.0000248\) meters has a mass of about \(4.79 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.238\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 41.2\).

The velocity after \(0.02\) seconds is approximately \(-0.134\) meters per second.

Example 84

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 84)

A water droplet with a radius of \(0.000224\) meters has a mass of about \(3.55 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.716\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.7\).

The velocity after \(0.03\) seconds is approximately \(-0.241\) meters per second.

Example 85

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 85)

A water droplet with a radius of \(0.0000821\) meters has a mass of about \(1.74 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.433\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 22.7\).

The velocity after \(0.03\) seconds is approximately \(-0.214\) meters per second.

Example 86

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 86)

A water droplet with a radius of \(0.0000477\) meters has a mass of about \(3.40 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.330\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.7\).

The velocity after \(0.02\) seconds is approximately \(-0.148\) meters per second.

Example 87

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 87)

A water droplet with a radius of \(0.000112\) meters has a mass of about \(4.37 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.505\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 19.4\).

The velocity after \(0.04\) seconds is approximately \(-0.273\) meters per second.

Example 88

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 88)

A water droplet with a radius of \(0.000268\) meters has a mass of about \(6.02 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.782\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 12.5\).

The velocity after \(0.02\) seconds is approximately \(-0.174\) meters per second.

Example 89

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 89)

A water droplet with a radius of \(0.000339\) meters has a mass of about \(1.22 \times 10^{-8}\) kilograms and a downward terminal velocity of approximately \(0.880\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 11.1\).

The velocity after \(0.02\) seconds is approximately \(-0.176\) meters per second.

Example 90

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 90)

A water droplet with a radius of \(0.000105\) meters has a mass of about \(3.60 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.489\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 20.1\).

The velocity after \(0.04\) seconds is approximately \(-0.270\) meters per second.

Example 91

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 91)

A water droplet with a radius of \(0.000224\) meters has a mass of about \(3.55 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.716\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.7\).

The velocity after \(0.02\) seconds is approximately \(-0.172\) meters per second.

Example 92

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 92)

A water droplet with a radius of \(0.0000329\) meters has a mass of about \(1.11 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.274\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 35.8\).

The velocity after \(0.04\) seconds is approximately \(-0.209\) meters per second.

Example 93

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 93)

A water droplet with a radius of \(0.0000145\) meters has a mass of about \(9.57 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.182\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 53.9\).

The velocity after \(0.03\) seconds is approximately \(-0.146\) meters per second.

Example 94

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 94)

A water droplet with a radius of \(0.000250\) meters has a mass of about \(4.92 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.756\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.0\).

The velocity after \(0.02\) seconds is approximately \(-0.173\) meters per second.

Example 95

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 95)

A water droplet with a radius of \(0.0000471\) meters has a mass of about \(3.28 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.328\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 29.9\).

The velocity after \(0.04\) seconds is approximately \(-0.229\) meters per second.

Example 96

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 96)

A water droplet with a radius of \(0.000237\) meters has a mass of about \(4.19 \times 10^{-9}\) kilograms and a downward terminal velocity of approximately \(0.736\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.04\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 13.3\).

The velocity after \(0.04\) seconds is approximately \(-0.304\) meters per second.

Example 97

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 97)

A water droplet with a radius of \(0.0000191\) meters has a mass of about \(2.19 \times 10^{-12}\) kilograms and a downward terminal velocity of approximately \(0.209\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 46.9\).

The velocity after \(0.03\) seconds is approximately \(-0.158\) meters per second.

Example 98

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 98)

A water droplet with a radius of \(0.0000434\) meters has a mass of about \(2.57 \times 10^{-11}\) kilograms and a downward terminal velocity of approximately \(0.315\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.02\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 31.1\).

The velocity after \(0.02\) seconds is approximately \(-0.146\) meters per second.

Example 99

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 99)

A water droplet with a radius of \(0.0000101\) meters has a mass of about \(3.25 \times 10^{-13}\) kilograms and a downward terminal velocity of approximately \(0.152\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 64.5\).

The velocity after \(0.03\) seconds is approximately \(-0.130\) meters per second.

Example 100

C3m - Model and analyze the vertical motion of an object with linear drag (ver. 100)

A water droplet with a radius of \(0.0000761\) meters has a mass of about \(1.38 \times 10^{-10}\) kilograms and a downward terminal velocity of approximately \(0.417\) meters per second.

Write an initial value problem (IVP) modeling the velocity of this water droplet when dropped from rest, assuming that the acceleration due to gravity is roughly \(9.81\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). Then solve this IVP to compute the droplet's velocity after \(0.03\) seconds.

Answer.

The IVP is given by

\[v'+Av=-g \hspace{3em} v(0)=0\]

where \(g=9.81\) and \(A\approx 23.5\).

The velocity after \(0.03\) seconds is approximately \(-0.211\) meters per second.