## 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 $$t=66000$$.

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.

## 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 $$t=25300$$.

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.

## 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 $$t=88200$$.

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.

## 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 $$t=30400$$.

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.

## 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 $$t=33600$$.

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.

## 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 $$t=72000$$.

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.

## 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 $$t=81900$$.

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.

## 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 $$t=110400$$.

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.

## 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 $$t=9000$$.

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.

## 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 $$t=42000$$.

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.

## 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 $$t=14400$$.

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.

## 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 $$t=48000$$.

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.

## 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 $$t=27000$$.

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.

## 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 $$t=8400$$.

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.

## 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 $$t=51000$$.

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.

## 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 $$t=19600$$.

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.

## 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 $$t=32200$$.

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.

## 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 $$t=41800$$.

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.

## 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 $$t=34300$$.

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.

## 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 $$t=61200$$.

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.

## 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 $$t=2100$$.

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.

## 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 $$t=52900$$.

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.

## 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 $$t=15200$$.

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.

## 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 $$t=90000$$.

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.

## 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 $$t=58500$$.

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.

## 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 $$t=72200$$.

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.

## 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 $$t=16400$$.

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.

## 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 $$t=39200$$.

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.

## 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 $$t=39200$$.

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.

## 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 $$t=58500$$.

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.

## 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 $$t=64000$$.

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.

## 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 $$t=36400$$.

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.

## 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 $$t=14700$$.

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.

## 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 $$t=103200$$.

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.

## 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 $$t=99000$$.

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.

## 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 $$t=15200$$.

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.

## 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 $$t=36000$$.

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.

## 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 $$t=9000$$.

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.

## 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 $$t=2200$$.

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.

## 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 $$t=24700$$.

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.

## 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 $$t=70000$$.

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.

## 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 $$t=50600$$.

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.

## 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 $$t=64800$$.

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.

## 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 $$t=2800$$.

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.

## 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 $$t=15000$$.

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.

## 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 $$t=80000$$.

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.

## 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 $$t=5400$$.

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.

## 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 $$t=52000$$.

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.

## 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 $$t=9600$$.

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.

## 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 $$t=3000$$.

$300 \, {v'} = 100 \, u\left({t} - 2400\right) - 100 \, u\left({t} - 3300\right)\hspace{2em}v(0)= 120$
${v} = \frac{1}{3} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{3} \, {\left({t} - 3300\right)} u\left({t} - 3300\right) + 120$
It follows that when $$t=3000$$, the velocity of the rocket is $$320$$ meters per second.