An Astronaut Adrift

Q. An astronaut accidentally gets separated out of his small spaceship accelerating in interstellar space at a constant rate of 100 m/s². Assume there are no nearby stars to exert gravitational force on him. What is the acceleration of the astronaut the instant after he is outside the spaceship, in m/s²? expects 0 m/s²

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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?

The playground

Inputs

ship acceleration

a_{s}

100 m/s²

Outputs

acceleration

a_{a}

The solution

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 Ruby

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

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

1require_relative "../solutions/an_astronaut_adrift"23play_with :an_astronaut_adrift do4  title "An Astronaut Adrift"5end

The laws

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)