A Rocket Lifts Off

solved NewtonsSecondLaw

Q. A rocket of 20000 kg $m = 20000$ kg at lift-off is blasted upward $F_{b} = | F_{b} |$ with an acceleration of 5.0 m/s² $a = 5.0$ m/s². Taking g as $W = - | W |$ 10 m/s² $g = 10$ m/s², what is the thrust of the blast, in newtons?

Inputs

mass

m

2e+04 kg

acceleration

a

5 m/s²

gravity

g

10 m/s²

Outputs

thrust

| F_{b} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 20000$ kg
$a = 5.0$ m/s²
$g = 10$ m/s²
$W = - | W |$ from “Taking g as”
$F_{b} = | F_{b} |$ from “blasted upward”

Find $| F_{b} |$ .

Working

second law(lifting off, y) $F_{net} = m \cdot a$
$F_{net} = 20000 \cdot 5$
$F_{net} = 100000$ N
weight of a body(rocket) $| W | = m \cdot g$
$| W | = 20000 \cdot 10$
$| W | = 200000$ N
Taking g as $W = - | W |$
$W = - 200000$
$W = −200000$ N
net force is the resultant(rocket, lifting off, y) $F_{net} = F_{b} + W$
$100000 = F_{b} + (−200000)$
$F_{b} = 300000$ N
magnitude from components(force on rocket by blast during lifting off) $| F_{b} | = \sqrt{{F_{b}}^{2}}$
$| F_{b} | = \sqrt{300000^{2}}$
$| F_{b} | = 300000$ N
$| F_{b} | \approx 3.0 \times 10^{5}$ Nto 2 significant figures

Conclusion

The blast has to hold the rocket up against its weight and still speed it upward, so it pushes with the mass times the pull and the acceleration together.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_rocket_lifts_off do4  kind :numerical5  describes "A rocket of %{mass} at lift-off is blasted upward with an acceleration of %{acceleration}. " \6            "Taking g as %{gravity}, %{thrust}, in newtons?"7  given mass: "20000 kg"8  given acceleration: "5.0 m/s2"9  given gravity: "10 m/s2"10  asks thrust: "what is the thrust of the blast"11end

solution.rb

1require_relative "../questions/a_rocket_lifts_off"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_rocket_lifts_off, using: NewtonsSecondLaw do5  objects :rocket, :blast6  axes :y7  rocket.goes(:lifting_off)8  deduce_from "Taking g as", rocket.weight.direction => -y910  take mass: rocket.mass11  take acceleration: rocket.acceleration12  take gravity: g1314  deduce_from "blasted upward", rocket.force(by: blast).direction => y1516  thrust = rocket.force(by: blast).magnitude17  answer thrust:, in: "N", figures: 21819  conclusion "The blast has to hold the rocket up against its weight and still speed it upward, so " \20             "it pushes with the mass times the pull and the acceleration together."21end

playground.rb

1require_relative "../solutions/a_rocket_lifts_off"23play_with :a_rocket_lifts_off do4  title "A Rocket Lifts Off"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$F$	N	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	$v_{av}$	m/s	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
gravity	$g$	m/s²	—	UniformAcceleration
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
displacement of rocket during lifting off along y	$Δ x$	m	—
position of rocket at end of lifting off along y	$x_{1}$	m	—
position of rocket at start of lifting off along y	$x_{0}$	m	—
average velocity of rocket during lifting off along y	$v_{av}$	m/s	—
time taken of lifting off	$Δ t$	s	0 … 1e+05
average speed of rocket during lifting off	$c_{av}$	m/s	—
path length of rocket during lifting off	$d$	m	—
change of velocity of rocket during lifting off along y	$Δ v$	m/s	—
velocity of rocket at end of lifting off from ground along y	$v_{1}$	m/s	—
velocity of rocket at start of lifting off from ground along y	$v_{0}$	m/s	—
average acceleration of rocket during lifting off along y	$a_{av}$	m/s²	—
acceleration of rocket during lifting off along y	$a$	m/s²	—
net force on rocket during lifting off along y	$F_{net}$	N	—
mass of rocket	$m$	kg	0 … ∞
mass of blast	$m_{b}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on rocket	$\| W \|$	N	—
weight on blast	$\| W_{b} \|$	N	—
force on rocket by blast during lifting off	$\| F_{b} \|$	N	—
weight on rocket along y	$W$	N	—
force on rocket by blast during lifting off along y	$F_{b}$	N	—
displacement of rocket during lifting off	$\| Δ x \|$	m	—
position of rocket at end of lifting off	$\| x_{1} \|$	m	—
position of rocket at start of lifting off	$\| x_{0} \|$	m	—
average velocity of rocket during lifting off	$\| v_{av} \|$	m/s	—
change of velocity of rocket during lifting off	$\| Δ v \|$	m/s	—
velocity of rocket at end of lifting off from ground	$\| v_{1} \|$	m/s	—
velocity of rocket at start of lifting off from ground	$\| v_{0} \|$	m/s	—
average acceleration of rocket during lifting off	$\| a_{av} \|$	m/s²	—
acceleration of rocket during lifting off	$\| a \|$	m/s²	—
net force on rocket during lifting off	$\| F_{net} \|$	N	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

v_{0} \equiv v_{i} (start of motion)

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

a_{i} (part 1) = a_{i} (whole)

steady through its parts

m_{b} \neq 0

has mass?

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y

Δ x = x_{1} - x_{0}

change of position(lifting off, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(lifting off, y)

c_{av} = \frac{d}{Δ t}

path a time(lifting off, y)

Δ v = v_{1} - v_{0}

velocity gained(lifting off, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(lifting off, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(lifting off, y)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(lifting off, y)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(lifting off, y)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(lifting off, y)

F_{net} = m \cdot a

second law(lifting off, y)

| W | = m \cdot g

weight of a body(rocket)

F_{net} = F_{b} + W

net force is the resultant(rocket, lifting off, y)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of rocket during lifting off)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of rocket at end of lifting off)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of rocket at start of lifting off)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of rocket during lifting off)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of rocket during lifting off)

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of rocket at end of lifting off from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of rocket at start of lifting off from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of rocket during lifting off)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of rocket during lifting off)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on rocket during lifting off)

| W | = \sqrt{W^{2}}

magnitude from components(weight on rocket)

| F_{b} | = \sqrt{{F_{b}}^{2}}

magnitude from components(force on rocket by blast during lifting off)

W = - | W |

Taking g as

F_{b} = | F_{b} |

blasted upward

m \neq 0

has mass?(rocket)

m_{b} \neq 0

has mass?(blast)

F_{net} = 0

free of forces?(rocket, lifting off)