D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration.


Example 1

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 1)

A rocket weighing \(3000\) kg is traveling at a constant \(170\) meters per second. Then when \(t=57000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(6000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=66000\).

Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 50 \, u\left({t} - 57000\right) - 50 \, u\left({t} - 63000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{60} \, {\left({t} - 57000\right)} u\left({t} - 57000\right) - \frac{1}{60} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) + 170\]

It follows that when \(t=66000\), the velocity of the rocket is \(270\) meters per second.


Example 2

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 2)

A rocket weighing \(2300\) kg is traveling at a constant \(200\) meters per second. Then when \(t=13800\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(6900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=25300\).

Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 20 \, u\left({t} - 13800\right) - 20 \, u\left({t} - 20700\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{1}{115} \, {\left({t} - 13800\right)} u\left({t} - 13800\right) - \frac{1}{115} \, {\left({t} - 20700\right)} u\left({t} - 20700\right) + 200\]

It follows that when \(t=25300\), the velocity of the rocket is \(260\) meters per second.


Example 3

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 3)

A rocket weighing \(4200\) kg is traveling at a constant \(90\) meters per second. Then when \(t=67200\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(12600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=88200\).

Answer.

An IVP modeling this scenario is given by:

\[4200 \, {v'} = 90 \, u\left({t} - 67200\right) - 90 \, u\left({t} - 79800\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{3}{140} \, {\left({t} - 67200\right)} u\left({t} - 67200\right) - \frac{3}{140} \, {\left({t} - 79800\right)} u\left({t} - 79800\right) + 90\]

It follows that when \(t=88200\), the velocity of the rocket is \(360\) meters per second.


Example 4

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 4)

A rocket weighing \(1600\) kg is traveling at a constant \(130\) meters per second. Then when \(t=28800\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=30400\).

Answer.

An IVP modeling this scenario is given by:

\[1600 \, {v'} = 20 \, u\left({t} - 28800\right) - 20 \, u\left({t} - 33600\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{1}{80} \, {\left({t} - 28800\right)} u\left({t} - 28800\right) - \frac{1}{80} \, {\left({t} - 33600\right)} u\left({t} - 33600\right) + 130\]

It follows that when \(t=30400\), the velocity of the rocket is \(150\) meters per second.


Example 5

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 5)

A rocket weighing \(2800\) kg is traveling at a constant \(10\) meters per second. Then when \(t=28000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(8400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=33600\).

Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 30 \, u\left({t} - 28000\right) - 30 \, u\left({t} - 36400\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{3}{280} \, {\left({t} - 28000\right)} u\left({t} - 28000\right) - \frac{3}{280} \, {\left({t} - 36400\right)} u\left({t} - 36400\right) + 10\]

It follows that when \(t=33600\), the velocity of the rocket is \(70\) meters per second.


Example 6

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 6)

A rocket weighing \(4500\) kg is traveling at a constant \(180\) meters per second. Then when \(t=63000\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(9000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=72000\).

Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 40 \, u\left({t} - 63000\right) - 40 \, u\left({t} - 72000\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{2}{225} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) - \frac{2}{225} \, {\left({t} - 72000\right)} u\left({t} - 72000\right) + 180\]

It follows that when \(t=72000\), the velocity of the rocket is \(260\) meters per second.


Example 7

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 7)

A rocket weighing \(3900\) kg is traveling at a constant \(170\) meters per second. Then when \(t=62400\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=81900\).

Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 60 \, u\left({t} - 62400\right) - 60 \, u\left({t} - 78000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{65} \, {\left({t} - 62400\right)} u\left({t} - 62400\right) - \frac{1}{65} \, {\left({t} - 78000\right)} u\left({t} - 78000\right) + 170\]

It follows that when \(t=81900\), the velocity of the rocket is \(410\) meters per second.


Example 8

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 8)

A rocket weighing \(4600\) kg is traveling at a constant \(180\) meters per second. Then when \(t=92000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=110400\).

Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 60 \, u\left({t} - 92000\right) - 60 \, u\left({t} - 101200\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{3}{230} \, {\left({t} - 92000\right)} u\left({t} - 92000\right) - \frac{3}{230} \, {\left({t} - 101200\right)} u\left({t} - 101200\right) + 180\]

It follows that when \(t=110400\), the velocity of the rocket is \(300\) meters per second.


Example 9

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 9)

A rocket weighing \(1500\) kg is traveling at a constant \(130\) meters per second. Then when \(t=6000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9000\).

Answer.

An IVP modeling this scenario is given by:

\[1500 \, {v'} = 60 \, u\left({t} - 6000\right) - 60 \, u\left({t} - 9000\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{1}{25} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) - \frac{1}{25} \, {\left({t} - 9000\right)} u\left({t} - 9000\right) + 130\]

It follows that when \(t=9000\), the velocity of the rocket is \(250\) meters per second.


Example 10

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 10)

A rocket weighing \(2100\) kg is traveling at a constant \(200\) meters per second. Then when \(t=39900\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(4200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=42000\).

Answer.

An IVP modeling this scenario is given by:

\[2100 \, {v'} = 50 \, u\left({t} - 39900\right) - 50 \, u\left({t} - 44100\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{1}{42} \, {\left({t} - 39900\right)} u\left({t} - 39900\right) - \frac{1}{42} \, {\left({t} - 44100\right)} u\left({t} - 44100\right) + 200\]

It follows that when \(t=42000\), the velocity of the rocket is \(250\) meters per second.


Example 11

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 11)

A rocket weighing \(2400\) kg is traveling at a constant \(100\) meters per second. Then when \(t=12000\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14400\).

Answer.

An IVP modeling this scenario is given by:

\[2400 \, {v'} = 20 \, u\left({t} - 12000\right) - 20 \, u\left({t} - 16800\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{120} \, {\left({t} - 12000\right)} u\left({t} - 12000\right) - \frac{1}{120} \, {\left({t} - 16800\right)} u\left({t} - 16800\right) + 100\]

It follows that when \(t=14400\), the velocity of the rocket is \(120\) meters per second.


Example 12

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 12)

A rocket weighing \(3000\) kg is traveling at a constant \(100\) meters per second. Then when \(t=24000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=48000\).

Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 50 \, u\left({t} - 24000\right) - 50 \, u\left({t} - 36000\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{60} \, {\left({t} - 24000\right)} u\left({t} - 24000\right) - \frac{1}{60} \, {\left({t} - 36000\right)} u\left({t} - 36000\right) + 100\]

It follows that when \(t=48000\), the velocity of the rocket is \(300\) meters per second.


Example 13

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 13)

A rocket weighing \(4500\) kg is traveling at a constant \(50\) meters per second. Then when \(t=13500\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(18000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=27000\).

Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 20 \, u\left({t} - 13500\right) - 20 \, u\left({t} - 31500\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{225} \, {\left({t} - 13500\right)} u\left({t} - 13500\right) - \frac{1}{225} \, {\left({t} - 31500\right)} u\left({t} - 31500\right) + 50\]

It follows that when \(t=27000\), the velocity of the rocket is \(110\) meters per second.


Example 14

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 14)

A rocket weighing \(700\) kg is traveling at a constant \(10\) meters per second. Then when \(t=6300\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(1400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=8400\).

Answer.

An IVP modeling this scenario is given by:

\[700 \, {v'} = 90 \, u\left({t} - 6300\right) - 90 \, u\left({t} - 7700\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{9}{70} \, {\left({t} - 6300\right)} u\left({t} - 6300\right) - \frac{9}{70} \, {\left({t} - 7700\right)} u\left({t} - 7700\right) + 10\]

It follows that when \(t=8400\), the velocity of the rocket is \(190\) meters per second.


Example 15

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 15)

A rocket weighing \(3400\) kg is traveling at a constant \(140\) meters per second. Then when \(t=44200\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(13600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=51000\).

Answer.

An IVP modeling this scenario is given by:

\[3400 \, {v'} = 20 \, u\left({t} - 44200\right) - 20 \, u\left({t} - 57800\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{1}{170} \, {\left({t} - 44200\right)} u\left({t} - 44200\right) - \frac{1}{170} \, {\left({t} - 57800\right)} u\left({t} - 57800\right) + 140\]

It follows that when \(t=51000\), the velocity of the rocket is \(180\) meters per second.


Example 16

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 16)

A rocket weighing \(2800\) kg is traveling at a constant \(90\) meters per second. Then when \(t=0\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(11200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=19600\).

Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = -10 \, u\left({t} - 11200\right) + 10 \, u\left({t}\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = -\frac{1}{280} \, {\left({t} - 11200\right)} u\left({t} - 11200\right) + \frac{1}{280} \, {t} u\left({t}\right) + 90\]

It follows that when \(t=19600\), the velocity of the rocket is \(130\) meters per second.


Example 17

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 17)

A rocket weighing \(2300\) kg is traveling at a constant \(100\) meters per second. Then when \(t=18400\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=32200\).

Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 50 \, u\left({t} - 18400\right) - 50 \, u\left({t} - 27600\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{46} \, {\left({t} - 18400\right)} u\left({t} - 18400\right) - \frac{1}{46} \, {\left({t} - 27600\right)} u\left({t} - 27600\right) + 100\]

It follows that when \(t=32200\), the velocity of the rocket is \(300\) meters per second.


Example 18

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 18)

A rocket weighing \(1900\) kg is traveling at a constant \(200\) meters per second. Then when \(t=38000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=41800\).

Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 60 \, u\left({t} - 38000\right) - 60 \, u\left({t} - 41800\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{3}{95} \, {\left({t} - 38000\right)} u\left({t} - 38000\right) - \frac{3}{95} \, {\left({t} - 41800\right)} u\left({t} - 41800\right) + 200\]

It follows that when \(t=41800\), the velocity of the rocket is \(320\) meters per second.


Example 19

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 19)

A rocket weighing \(4900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=24500\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(14700\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=34300\).

Answer.

An IVP modeling this scenario is given by:

\[4900 \, {v'} = 100 \, u\left({t} - 24500\right) - 100 \, u\left({t} - 39200\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{49} \, {\left({t} - 24500\right)} u\left({t} - 24500\right) - \frac{1}{49} \, {\left({t} - 39200\right)} u\left({t} - 39200\right) + 80\]

It follows that when \(t=34300\), the velocity of the rocket is \(280\) meters per second.


Example 20

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 20)

A rocket weighing \(3400\) kg is traveling at a constant \(90\) meters per second. Then when \(t=57800\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(13600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=61200\).

Answer.

An IVP modeling this scenario is given by:

\[3400 \, {v'} = 60 \, u\left({t} - 57800\right) - 60 \, u\left({t} - 71400\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{3}{170} \, {\left({t} - 57800\right)} u\left({t} - 57800\right) - \frac{3}{170} \, {\left({t} - 71400\right)} u\left({t} - 71400\right) + 90\]

It follows that when \(t=61200\), the velocity of the rocket is \(150\) meters per second.


Example 21

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 21)

A rocket weighing \(100\) kg is traveling at a constant \(50\) meters per second. Then when \(t=1900\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2100\).

Answer.

An IVP modeling this scenario is given by:

\[100 \, {v'} = 30 \, u\left({t} - 1900\right) - 30 \, u\left({t} - 2200\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{3}{10} \, {\left({t} - 1900\right)} u\left({t} - 1900\right) - \frac{3}{10} \, {\left({t} - 2200\right)} u\left({t} - 2200\right) + 50\]

It follows that when \(t=2100\), the velocity of the rocket is \(110\) meters per second.


Example 22

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 22)

A rocket weighing \(2300\) kg is traveling at a constant \(60\) meters per second. Then when \(t=43700\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=52900\).

Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 20 \, u\left({t} - 43700\right) - 20 \, u\left({t} - 48300\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{115} \, {\left({t} - 43700\right)} u\left({t} - 43700\right) - \frac{1}{115} \, {\left({t} - 48300\right)} u\left({t} - 48300\right) + 60\]

It follows that when \(t=52900\), the velocity of the rocket is \(100\) meters per second.


Example 23

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 23)

A rocket weighing \(800\) kg is traveling at a constant \(20\) meters per second. Then when \(t=14400\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(2400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15200\).

Answer.

An IVP modeling this scenario is given by:

\[800 \, {v'} = 20 \, u\left({t} - 14400\right) - 20 \, u\left({t} - 16800\right)\hspace{2em}v(0)= 20\]

This IVP solves to:

\[{v} = \frac{1}{40} \, {\left({t} - 14400\right)} u\left({t} - 14400\right) - \frac{1}{40} \, {\left({t} - 16800\right)} u\left({t} - 16800\right) + 20\]

It follows that when \(t=15200\), the velocity of the rocket is \(40\) meters per second.


Example 24

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 24)

A rocket weighing \(4500\) kg is traveling at a constant \(90\) meters per second. Then when \(t=76500\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(13500\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=90000\).

Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 40 \, u\left({t} - 76500\right) - 40 \, u\left({t} - 90000\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{2}{225} \, {\left({t} - 76500\right)} u\left({t} - 76500\right) - \frac{2}{225} \, {\left({t} - 90000\right)} u\left({t} - 90000\right) + 90\]

It follows that when \(t=90000\), the velocity of the rocket is \(210\) meters per second.


Example 25

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 25)

A rocket weighing \(3900\) kg is traveling at a constant \(100\) meters per second. Then when \(t=27300\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=58500\).

Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 100 \, u\left({t} - 27300\right) - 100 \, u\left({t} - 42900\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{39} \, {\left({t} - 27300\right)} u\left({t} - 27300\right) - \frac{1}{39} \, {\left({t} - 42900\right)} u\left({t} - 42900\right) + 100\]

It follows that when \(t=58500\), the velocity of the rocket is \(500\) meters per second.


Example 26

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 26)

A rocket weighing \(3800\) kg is traveling at a constant \(190\) meters per second. Then when \(t=41800\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(15200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=72200\).

Answer.

An IVP modeling this scenario is given by:

\[3800 \, {v'} = 10 \, u\left({t} - 41800\right) - 10 \, u\left({t} - 57000\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{1}{380} \, {\left({t} - 41800\right)} u\left({t} - 41800\right) - \frac{1}{380} \, {\left({t} - 57000\right)} u\left({t} - 57000\right) + 190\]

It follows that when \(t=72200\), the velocity of the rocket is \(230\) meters per second.


Example 27

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 27)

A rocket weighing \(4100\) kg is traveling at a constant \(40\) meters per second. Then when \(t=4100\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(8200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=16400\).

Answer.

An IVP modeling this scenario is given by:

\[4100 \, {v'} = 100 \, u\left({t} - 4100\right) - 100 \, u\left({t} - 12300\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{1}{41} \, {\left({t} - 4100\right)} u\left({t} - 4100\right) - \frac{1}{41} \, {\left({t} - 12300\right)} u\left({t} - 12300\right) + 40\]

It follows that when \(t=16400\), the velocity of the rocket is \(240\) meters per second.


Example 28

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 28)

A rocket weighing \(4900\) kg is traveling at a constant \(90\) meters per second. Then when \(t=19600\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(9800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=39200\).

Answer.

An IVP modeling this scenario is given by:

\[4900 \, {v'} = 90 \, u\left({t} - 19600\right) - 90 \, u\left({t} - 29400\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{9}{490} \, {\left({t} - 19600\right)} u\left({t} - 19600\right) - \frac{9}{490} \, {\left({t} - 29400\right)} u\left({t} - 29400\right) + 90\]

It follows that when \(t=39200\), the velocity of the rocket is \(270\) meters per second.


Example 29

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 29)

A rocket weighing \(2800\) kg is traveling at a constant \(170\) meters per second. Then when \(t=30800\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(5600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=39200\).

Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 40 \, u\left({t} - 30800\right) - 40 \, u\left({t} - 36400\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{70} \, {\left({t} - 30800\right)} u\left({t} - 30800\right) - \frac{1}{70} \, {\left({t} - 36400\right)} u\left({t} - 36400\right) + 170\]

It follows that when \(t=39200\), the velocity of the rocket is \(250\) meters per second.


Example 30

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 30)

A rocket weighing \(3900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=54600\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=58500\).

Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 10 \, u\left({t} - 54600\right) - 10 \, u\left({t} - 70200\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{390} \, {\left({t} - 54600\right)} u\left({t} - 54600\right) - \frac{1}{390} \, {\left({t} - 70200\right)} u\left({t} - 70200\right) + 80\]

It follows that when \(t=58500\), the velocity of the rocket is \(90\) meters per second.


Example 31

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 31)

A rocket weighing \(3200\) kg is traveling at a constant \(10\) meters per second. Then when \(t=57600\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(6400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=64000\).

Answer.

An IVP modeling this scenario is given by:

\[3200 \, {v'} = 70 \, u\left({t} - 57600\right) - 70 \, u\left({t} - 64000\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{7}{320} \, {\left({t} - 57600\right)} u\left({t} - 57600\right) - \frac{7}{320} \, {\left({t} - 64000\right)} u\left({t} - 64000\right) + 10\]

It follows that when \(t=64000\), the velocity of the rocket is \(150\) meters per second.


Example 32

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 32)

A rocket weighing \(2800\) kg is traveling at a constant \(110\) meters per second. Then when \(t=14000\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(11200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=36400\).

Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 80 \, u\left({t} - 14000\right) - 80 \, u\left({t} - 25200\right)\hspace{2em}v(0)= 110\]

This IVP solves to:

\[{v} = \frac{1}{35} \, {\left({t} - 14000\right)} u\left({t} - 14000\right) - \frac{1}{35} \, {\left({t} - 25200\right)} u\left({t} - 25200\right) + 110\]

It follows that when \(t=36400\), the velocity of the rocket is \(430\) meters per second.


Example 33

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 33)

A rocket weighing \(2100\) kg is traveling at a constant \(10\) meters per second. Then when \(t=2100\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(8400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14700\).

Answer.

An IVP modeling this scenario is given by:

\[2100 \, {v'} = 90 \, u\left({t} - 2100\right) - 90 \, u\left({t} - 10500\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{3}{70} \, {\left({t} - 2100\right)} u\left({t} - 2100\right) - \frac{3}{70} \, {\left({t} - 10500\right)} u\left({t} - 10500\right) + 10\]

It follows that when \(t=14700\), the velocity of the rocket is \(370\) meters per second.


Example 34

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 34)

A rocket weighing \(4300\) kg is traveling at a constant \(130\) meters per second. Then when \(t=77400\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(12900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=103200\).

Answer.

An IVP modeling this scenario is given by:

\[4300 \, {v'} = 40 \, u\left({t} - 77400\right) - 40 \, u\left({t} - 90300\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{2}{215} \, {\left({t} - 77400\right)} u\left({t} - 77400\right) - \frac{2}{215} \, {\left({t} - 90300\right)} u\left({t} - 90300\right) + 130\]

It follows that when \(t=103200\), the velocity of the rocket is \(250\) meters per second.


Example 35

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 35)

A rocket weighing \(4500\) kg is traveling at a constant \(60\) meters per second. Then when \(t=63000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(18000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=99000\).

Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 60 \, u\left({t} - 63000\right) - 60 \, u\left({t} - 81000\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{75} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) - \frac{1}{75} \, {\left({t} - 81000\right)} u\left({t} - 81000\right) + 60\]

It follows that when \(t=99000\), the velocity of the rocket is \(300\) meters per second.


Example 36

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 36)

A rocket weighing \(1900\) kg is traveling at a constant \(170\) meters per second. Then when \(t=13300\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15200\).

Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 60 \, u\left({t} - 13300\right) - 60 \, u\left({t} - 17100\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{3}{95} \, {\left({t} - 13300\right)} u\left({t} - 13300\right) - \frac{3}{95} \, {\left({t} - 17100\right)} u\left({t} - 17100\right) + 170\]

It follows that when \(t=15200\), the velocity of the rocket is \(230\) meters per second.


Example 37

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 37)

A rocket weighing \(3600\) kg is traveling at a constant \(160\) meters per second. Then when \(t=25200\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(7200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=36000\).

Answer.

An IVP modeling this scenario is given by:

\[3600 \, {v'} = 100 \, u\left({t} - 25200\right) - 100 \, u\left({t} - 32400\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{1}{36} \, {\left({t} - 25200\right)} u\left({t} - 25200\right) - \frac{1}{36} \, {\left({t} - 32400\right)} u\left({t} - 32400\right) + 160\]

It follows that when \(t=36000\), the velocity of the rocket is \(360\) meters per second.


Example 38

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 38)

A rocket weighing \(1000\) kg is traveling at a constant \(50\) meters per second. Then when \(t=1000\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(4000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9000\).

Answer.

An IVP modeling this scenario is given by:

\[1000 \, {v'} = 10 \, u\left({t} - 1000\right) - 10 \, u\left({t} - 5000\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{100} \, {\left({t} - 1000\right)} u\left({t} - 1000\right) - \frac{1}{100} \, {\left({t} - 5000\right)} u\left({t} - 5000\right) + 50\]

It follows that when \(t=9000\), the velocity of the rocket is \(90\) meters per second.


Example 39

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 39)

A rocket weighing \(1100\) kg is traveling at a constant \(170\) meters per second. Then when \(t=0\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(3300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2200\).

Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = -70 \, u\left({t} - 3300\right) + 70 \, u\left({t}\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = -\frac{7}{110} \, {\left({t} - 3300\right)} u\left({t} - 3300\right) + \frac{7}{110} \, {t} u\left({t}\right) + 170\]

It follows that when \(t=2200\), the velocity of the rocket is \(310\) meters per second.


Example 40

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 40)

A rocket weighing \(1900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=19000\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=24700\).

Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 20 \, u\left({t} - 19000\right) - 20 \, u\left({t} - 22800\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{95} \, {\left({t} - 19000\right)} u\left({t} - 19000\right) - \frac{1}{95} \, {\left({t} - 22800\right)} u\left({t} - 22800\right) + 80\]

It follows that when \(t=24700\), the velocity of the rocket is \(120\) meters per second.


Example 41

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 41)

A rocket weighing \(5000\) kg is traveling at a constant \(70\) meters per second. Then when \(t=50000\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(10000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=70000\).

Answer.

An IVP modeling this scenario is given by:

\[5000 \, {v'} = 100 \, u\left({t} - 50000\right) - 100 \, u\left({t} - 60000\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{1}{50} \, {\left({t} - 50000\right)} u\left({t} - 50000\right) - \frac{1}{50} \, {\left({t} - 60000\right)} u\left({t} - 60000\right) + 70\]

It follows that when \(t=70000\), the velocity of the rocket is \(270\) meters per second.


Example 42

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 42)

A rocket weighing \(4600\) kg is traveling at a constant \(80\) meters per second. Then when \(t=32200\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(18400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=50600\).

Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 80 \, u\left({t} - 32200\right) - 80 \, u\left({t} - 50600\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{2}{115} \, {\left({t} - 32200\right)} u\left({t} - 32200\right) - \frac{2}{115} \, {\left({t} - 50600\right)} u\left({t} - 50600\right) + 80\]

It follows that when \(t=50600\), the velocity of the rocket is \(400\) meters per second.


Example 43

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 43)

A rocket weighing \(3600\) kg is traveling at a constant \(50\) meters per second. Then when \(t=39600\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(14400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=64800\).

Answer.

An IVP modeling this scenario is given by:

\[3600 \, {v'} = 50 \, u\left({t} - 39600\right) - 50 \, u\left({t} - 54000\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{72} \, {\left({t} - 39600\right)} u\left({t} - 39600\right) - \frac{1}{72} \, {\left({t} - 54000\right)} u\left({t} - 54000\right) + 50\]

It follows that when \(t=64800\), the velocity of the rocket is \(250\) meters per second.


Example 44

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 44)

A rocket weighing \(200\) kg is traveling at a constant \(10\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2800\).

Answer.

An IVP modeling this scenario is given by:

\[200 \, {v'} = 10 \, u\left({t} - 2400\right) - 10 \, u\left({t} - 2800\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{1}{20} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{20} \, {\left({t} - 2800\right)} u\left({t} - 2800\right) + 10\]

It follows that when \(t=2800\), the velocity of the rocket is \(30\) meters per second.


Example 45

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 45)

A rocket weighing \(3000\) kg is traveling at a constant \(170\) meters per second. Then when \(t=6000\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15000\).

Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 10 \, u\left({t} - 6000\right) - 10 \, u\left({t} - 18000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{300} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) - \frac{1}{300} \, {\left({t} - 18000\right)} u\left({t} - 18000\right) + 170\]

It follows that when \(t=15000\), the velocity of the rocket is \(200\) meters per second.


Example 46

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 46)

A rocket weighing \(4000\) kg is traveling at a constant \(70\) meters per second. Then when \(t=76000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=80000\).

Answer.

An IVP modeling this scenario is given by:

\[4000 \, {v'} = 50 \, u\left({t} - 76000\right) - 50 \, u\left({t} - 88000\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{1}{80} \, {\left({t} - 76000\right)} u\left({t} - 76000\right) - \frac{1}{80} \, {\left({t} - 88000\right)} u\left({t} - 88000\right) + 70\]

It follows that when \(t=80000\), the velocity of the rocket is \(120\) meters per second.


Example 47

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 47)

A rocket weighing \(1800\) kg is traveling at a constant \(170\) meters per second. Then when \(t=1800\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(3600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=5400\).

Answer.

An IVP modeling this scenario is given by:

\[1800 \, {v'} = 10 \, u\left({t} - 1800\right) - 10 \, u\left({t} - 5400\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{180} \, {\left({t} - 1800\right)} u\left({t} - 1800\right) - \frac{1}{180} \, {\left({t} - 5400\right)} u\left({t} - 5400\right) + 170\]

It follows that when \(t=5400\), the velocity of the rocket is \(190\) meters per second.


Example 48

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 48)

A rocket weighing \(2600\) kg is traveling at a constant \(40\) meters per second. Then when \(t=39000\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(10400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=52000\).

Answer.

An IVP modeling this scenario is given by:

\[2600 \, {v'} = 70 \, u\left({t} - 39000\right) - 70 \, u\left({t} - 49400\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{7}{260} \, {\left({t} - 39000\right)} u\left({t} - 39000\right) - \frac{7}{260} \, {\left({t} - 49400\right)} u\left({t} - 49400\right) + 40\]

It follows that when \(t=52000\), the velocity of the rocket is \(320\) meters per second.


Example 49

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 49)

A rocket weighing \(1200\) kg is traveling at a constant \(100\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(3600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9600\).

Answer.

An IVP modeling this scenario is given by:

\[1200 \, {v'} = 30 \, u\left({t} - 2400\right) - 30 \, u\left({t} - 6000\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{40} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{40} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) + 100\]

It follows that when \(t=9600\), the velocity of the rocket is \(190\) meters per second.


Example 50

D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 50)

A rocket weighing \(300\) kg is traveling at a constant \(120\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=3000\).

Answer.

An IVP modeling this scenario is given by:

\[300 \, {v'} = 100 \, u\left({t} - 2400\right) - 100 \, u\left({t} - 3300\right)\hspace{2em}v(0)= 120\]

This IVP solves to:

\[{v} = \frac{1}{3} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{3} \, {\left({t} - 3300\right)} u\left({t} - 3300\right) + 120\]

It follows that when \(t=3000\), the velocity of the rocket is \(320\) meters per second.