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 \(t=66000\).

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.

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 \(t=25300\).

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.

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 \(t=88200\).

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.

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 \(t=30400\).

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.

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 \(t=33600\).

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.

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 \(t=72000\).

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.

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 \(t=81900\).

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.

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 \(t=110400\).

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.

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 \(t=9000\).

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.

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 \(t=42000\).

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.

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 \(t=14400\).

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.

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 \(t=48000\).

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.

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 \(t=27000\).

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.

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 \(t=8400\).

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.

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 \(t=51000\).

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.

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 \(t=19600\).

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.

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 \(t=32200\).

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.

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 \(t=41800\).

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.

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 \(t=34300\).

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.

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 \(t=61200\).

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.

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 \(t=2100\).

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.

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 \(t=52900\).

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.

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 \(t=15200\).

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.

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 \(t=90000\).

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.

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 \(t=58500\).

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.

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 \(t=72200\).

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.

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 \(t=16400\).

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.

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 \(t=39200\).

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.

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 \(t=39200\).

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.

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 \(t=58500\).

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.

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 \(t=64000\).

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.

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 \(t=36400\).

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.

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 \(t=14700\).

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.

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 \(t=103200\).

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.

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 \(t=99000\).

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.

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 \(t=15200\).

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.

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 \(t=36000\).

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.

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 \(t=9000\).

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.

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 \(t=2200\).

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.

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 \(t=24700\).

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.

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 \(t=70000\).

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.

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 \(t=50600\).

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.

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 \(t=64800\).

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.

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 \(t=2800\).

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.

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 \(t=15000\).

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.

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 \(t=80000\).

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.

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 \(t=5400\).

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.

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 \(t=52000\).

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.

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 \(t=9600\).

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.

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 \(t=3000\).

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.