An Astronaut Adrift

solved NewtonsSecondLaw

Q. An astronaut accidentally gets separated out of his small spaceship accelerating in interstellar space at a constant rate of 100 m/s² $a_{s} (s) = 100$ m/s². Assume there are no nearby stars $F_{net,a} (a) = 0$ N to exert gravitational force on him. What is the acceleration of the astronaut the instant after he is outside the spaceship, in m/s²?

Inputs

ship acceleration

a_{s}

100 m/s²

Outputs

acceleration

a_{a}

How it's solved

Solved with NewtonsSecondLaw

Given

$a_{s} (s) = 100$ m/s²
$F_{net,a} (a) = 0$ N as it is free of forces, from “no nearby stars”

Find $a_{a} (a)$ .

Working

second law(adrift, x) $F_{net,a} (a) = m_{a} \cdot a_{a} (a)$
$a_{a} (a) = \frac{0}{m_{a}}$ since $m_{a} \neq 0$
$a_{a} (a) = 0$ m/s²

Conclusion

Once he is outside, nothing pushes or pulls him, so by the first law he stops speeding up at once; the ship's acceleration stays with the ship.

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 :an_astronaut_adrift do4  kind :numerical5  describes "An astronaut accidentally gets separated out of his small spaceship accelerating in interstellar space " \6            "at a constant rate of %{ship_acceleration}. Assume there are no nearby stars to exert gravitational " \7            "force on him. %{acceleration}, in m/s²?"8  given ship_acceleration: "100 m/s2"9  asks acceleration: "What is the acceleration of the astronaut the instant after he is outside the spaceship"10end

solution.rb

1require_relative "../questions/an_astronaut_adrift"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :an_astronaut_adrift, using: NewtonsSecondLaw do5  objects :astronaut, :spaceship6  axes :x7  spaceship.goes(:speeding_up)8  astronaut.goes(:adrift)910  take ship_acceleration: spaceship.acceleration1112  deduce_from "no nearby stars", astronaut.free_of_forces!1314  acceleration = astronaut.acceleration15  answer acceleration:, in: "m/s2"1617  conclusion "Once he is outside, nothing pushes or pulls him, so by the first law he stops speeding " \18             "up at once; the ship's acceleration stays with the ship."19end

playground.rb

1require_relative "../solutions/an_astronaut_adrift"23play_with :an_astronaut_adrift do4  title "An Astronaut Adrift"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 spaceship during speeding up along x	$Δ x_{s} (s)$	m	—
position of spaceship at end of speeding up along x	$x_{s} (s₁)$	m	—
position of spaceship at start of speeding up along x	$x_{s} (s₀)$	m	—
displacement of astronaut during adrift along x	$Δ x_{a} (a)$	m	—
position of astronaut at end of adrift along x	$x_{a} (a₁)$	m	—
position of astronaut at start of adrift along x	$x_{a} (a₀)$	m	—
average velocity of spaceship during speeding up along x	$v_{av,s} (s)$	m/s	—
time taken of speeding up	$Δ t_{s}$	s	0 … 1e+05
average velocity of astronaut during adrift along x	$v_{av,a} (a)$	m/s	—
time taken of adrift	$Δ t_{a}$	s	0 … 1e+05
average speed of spaceship during speeding up	$c_{av,s} (s)$	m/s	—
path length of spaceship during speeding up	$d_{s} (s)$	m	—
average speed of astronaut during adrift	$c_{av,a} (a)$	m/s	—
path length of astronaut during adrift	$d_{a} (a)$	m	—
change of velocity of spaceship during speeding up along x	$Δ v_{s} (s)$	m/s	—
velocity of spaceship at end of speeding up from ground along x	$v_{s} (s₁)$	m/s	—
velocity of spaceship at start of speeding up from ground along x	$v_{s} (s₀)$	m/s	—
change of velocity of astronaut during adrift along x	$Δ v_{a} (a)$	m/s	—
velocity of astronaut at end of adrift from ground along x	$v_{a} (a₁)$	m/s	—
velocity of astronaut at start of adrift from ground along x	$v_{a} (a₀)$	m/s	—
average acceleration of spaceship during speeding up along x	$a_{av,s} (s)$	m/s²	—
average acceleration of astronaut during adrift along x	$a_{av,a} (a)$	m/s²	—
acceleration of spaceship during speeding up along x	$a_{s} (s)$	m/s²	—
acceleration of astronaut during adrift along x	$a_{a} (a)$	m/s²	—
net force on spaceship during speeding up along x	$F_{net,s} (s)$	N	—
mass of spaceship	$m_{s}$	kg	0 … ∞
net force on astronaut during adrift along x	$F_{net,a} (a)$	N	—
mass of astronaut	$m_{a}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on astronaut	$\| W_{a} \|$	N	—
weight on spaceship	$\| W_{s} \|$	N	—
displacement of spaceship during speeding up	$\| Δ x_{s} (s) \|$	m	—
position of spaceship at end of speeding up	$\| x_{s} (s₁) \|$	m	—
position of spaceship at start of speeding up	$\| x_{s} (s₀) \|$	m	—
displacement of astronaut during adrift	$\| Δ x_{a} (a) \|$	m	—
position of astronaut at end of adrift	$\| x_{a} (a₁) \|$	m	—
position of astronaut at start of adrift	$\| x_{a} (a₀) \|$	m	—
average velocity of spaceship during speeding up	$\| v_{av,s} (s) \|$	m/s	—
average velocity of astronaut during adrift	$\| v_{av,a} (a) \|$	m/s	—
change of velocity of spaceship during speeding up	$\| Δ v_{s} (s) \|$	m/s	—
velocity of spaceship at end of speeding up from ground	$\| v_{s} (s₁) \|$	m/s	—
velocity of spaceship at start of speeding up from ground	$\| v_{s} (s₀) \|$	m/s	—
change of velocity of astronaut during adrift	$\| Δ v_{a} (a) \|$	m/s	—
velocity of astronaut at end of adrift from ground	$\| v_{a} (a₁) \|$	m/s	—
velocity of astronaut at start of adrift from ground	$\| v_{a} (a₀) \|$	m/s	—
average acceleration of spaceship during speeding up	$\| a_{av,s} (s) \|$	m/s²	—
average acceleration of astronaut during adrift	$\| a_{av,a} (a) \|$	m/s²	—
acceleration of spaceship during speeding up	$\| a_{s} (s) \|$	m/s²	—
acceleration of astronaut during adrift	$\| a_{a} (a) \|$	m/s²	—
net force on spaceship during speeding up	$\| F_{net,s} (s) \|$	N	—
net force on astronaut during adrift	$\| F_{net,a} (a) \|$	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_{s} (s) = x_{s} (s₁) - x_{s} (s₀)

change of position(speeding up, x)

Δ x_{a} (a) = x_{a} (a₁) - x_{a} (a₀)

change of position(adrift, x)

v_{av,s} (s) = \frac{Δ x_{s} (s)}{Δ t_{s}}

displacement a time(speeding up, x)

v_{av,a} (a) = \frac{Δ x_{a} (a)}{Δ t_{a}}

displacement a time(adrift, x)

c_{av,s} (s) = \frac{d_{s} (s)}{Δ t_{s}}

path a time(speeding up, x)

c_{av,a} (a) = \frac{d_{a} (a)}{Δ t_{a}}

path a time(adrift, x)

Δ v_{s} (s) = v_{s} (s₁) - v_{s} (s₀)

velocity gained(speeding up, x)

Δ v_{a} (a) = v_{a} (a₁) - v_{a} (a₀)

velocity gained(adrift, x)

a_{av,s} (s) = \frac{Δ v_{s} (s)}{Δ t_{s}}

velocity gained a time(speeding up, x)

a_{av,a} (a) = \frac{Δ v_{a} (a)}{Δ t_{a}}

velocity gained a time(adrift, x)

v_{s} (s₁) = v_{s} (s₀) + a_{s} (s) \cdot Δ t_{s}

velocity after a time(speeding up, x)

v_{a} (a₁) = v_{a} (a₀) + a_{a} (a) \cdot Δ t_{a}

velocity after a time(adrift, x)

x_{s} (s₁) = x_{s} (s₀) + v_{s} (s₀) \cdot Δ t_{s} + \frac{a_{s} (s) \cdot {Δ t_{s}}^{2}}{2}

position after a time(speeding up, x)

x_{a} (a₁) = x_{a} (a₀) + v_{a} (a₀) \cdot Δ t_{a} + \frac{a_{a} (a) \cdot {Δ t_{a}}^{2}}{2}

position after a time(adrift, x)

{v_{s} (s₁)}^{2} = {v_{s} (s₀)}^{2} + 2 \cdot a_{s} (s) \cdot (x_{s} (s₁) - x_{s} (s₀))

velocity after a distance(speeding up, x)

{v_{a} (a₁)}^{2} = {v_{a} (a₀)}^{2} + 2 \cdot a_{a} (a) \cdot (x_{a} (a₁) - x_{a} (a₀))

velocity after a distance(adrift, x)

x_{s} (s₁) - x_{s} (s₀) = \frac{(v_{s} (s₁) + v_{s} (s₀)) \cdot Δ t_{s}}{2}

mean of the velocities(speeding up, x)

x_{a} (a₁) - x_{a} (a₀) = \frac{(v_{a} (a₁) + v_{a} (a₀)) \cdot Δ t_{a}}{2}

mean of the velocities(adrift, x)

F_{net,s} (s) = m_{s} \cdot a_{s} (s)

second law(speeding up, x)

F_{net,a} (a) = m_{a} \cdot a_{a} (a)

second law(adrift, x)

| Δ x_{s} (s) | = \sqrt{{Δ x_{s} (s)}^{2}}

magnitude from components(displacement of spaceship during speeding up)

| x_{s} (s₁) | = \sqrt{{x_{s} (s₁)}^{2}}

magnitude from components(position of spaceship at end of speeding up)

| x_{s} (s₀) | = \sqrt{{x_{s} (s₀)}^{2}}

magnitude from components(position of spaceship at start of speeding up)

| Δ x_{a} (a) | = \sqrt{{Δ x_{a} (a)}^{2}}

magnitude from components(displacement of astronaut during adrift)

| x_{a} (a₁) | = \sqrt{{x_{a} (a₁)}^{2}}

magnitude from components(position of astronaut at end of adrift)

| x_{a} (a₀) | = \sqrt{{x_{a} (a₀)}^{2}}

magnitude from components(position of astronaut at start of adrift)

| v_{av,s} (s) | = \sqrt{{v_{av,s} (s)}^{2}}

magnitude from components(average velocity of spaceship during speeding up)

| v_{av,a} (a) | = \sqrt{{v_{av,a} (a)}^{2}}

magnitude from components(average velocity of astronaut during adrift)

| Δ v_{s} (s) | = \sqrt{{Δ v_{s} (s)}^{2}}

magnitude from components(change of velocity of spaceship during speeding up)

| v_{s} (s₁) | = \sqrt{{v_{s} (s₁)}^{2}}

magnitude from components(velocity of spaceship at end of speeding up from ground)

| v_{s} (s₀) | = \sqrt{{v_{s} (s₀)}^{2}}

magnitude from components(velocity of spaceship at start of speeding up from ground)

| Δ v_{a} (a) | = \sqrt{{Δ v_{a} (a)}^{2}}

magnitude from components(change of velocity of astronaut during adrift)

| v_{a} (a₁) | = \sqrt{{v_{a} (a₁)}^{2}}

magnitude from components(velocity of astronaut at end of adrift from ground)

| v_{a} (a₀) | = \sqrt{{v_{a} (a₀)}^{2}}

magnitude from components(velocity of astronaut at start of adrift from ground)

| a_{av,s} (s) | = \sqrt{{a_{av,s} (s)}^{2}}

magnitude from components(average acceleration of spaceship during speeding up)

| a_{av,a} (a) | = \sqrt{{a_{av,a} (a)}^{2}}

magnitude from components(average acceleration of astronaut during adrift)

| a_{s} (s) | = \sqrt{{a_{s} (s)}^{2}}

magnitude from components(acceleration of spaceship during speeding up)

| a_{a} (a) | = \sqrt{{a_{a} (a)}^{2}}

magnitude from components(acceleration of astronaut during adrift)

| F_{net,s} (s) | = \sqrt{{F_{net,s} (s)}^{2}}

magnitude from components(net force on spaceship during speeding up)

| F_{net,a} (a) | = \sqrt{{F_{net,a} (a)}^{2}}

magnitude from components(net force on astronaut during adrift)

m_{a} \neq 0

has mass?(astronaut)

m_{s} \neq 0

has mass?(spaceship)

F_{net,a} (a) = 0

free of forces?(astronaut, adrift)

F_{net,s} (s) = 0

free of forces?(spaceship, speeding up)