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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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 \(9.8\hspace{0.3em}\mathrm{m}/\mathrm{s}^2\). 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.