A Satellite Rocketed Free

solved EarthSatellites

Q. A satellite orbits the earth at a height of 400 km $h_{o} = 400$ km above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite = 200 kg $m = 200$ kg; mass of the earth = 6.0 × 10²⁴ kg $M_{o} = 6.0 \times 10^{24}$ kg; radius of the earth = 6.4 × 10⁶ m $R_{c,o} = 6.4 \times 10^{6}$ m. Answer in joules.

Inputs

height

h_{o}

400 km

satellite

m

200 kg

earth

M_{o}

6e+24 kg

radius

R_{c,o}

6400000.00 m

Outputs

energy

E_{b}

How it's solved

Solved with EarthSatellites

Given

$h_{o} = 400$ km
$m = 200$ kg
$M_{o} = 6.0 \times 10^{24}$ kg
$R_{c,o} = 6.4 \times 10^{6}$ m

Find $E_{b}$ .

Working

$h_{o} = 400$ km $= 400000$ m
orbit radius from the centre(round) $r_{o} = R_{c,o} + h_{o}$
$r_{o} = 6.4 \times 10^{6} + 400000$
$r_{o} = 6.8 \times 10^{6}$ m
binding energy of an orbit(round) $E_{b} = \frac{G \cdot M_{o} \cdot m}{2 \cdot r_{o}}$
$E_{b} = \frac{6.674 \times 10^{−11} \cdot 6 \times 10^{24} \cdot 200}{2 \cdot 6.8 \times 10^{6}}$
$E_{b} = 5.889 \times 10^{9}$ J
$E_{b} \approx 5.9 \times 10^{9}$ Jto 2 significant figures

Conclusion

In orbit the satellite's kinetic energy makes up half of its negative potential energy, so it is still bound; freeing it takes the other half.

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_satellite_rocketed_free do4  kind :numerical5  describes "A satellite orbits the earth at a height of %{height} above the surface. %{energy}? Mass of the " \6            "satellite = %{satellite}; mass of the earth = %{earth}; radius of the earth = %{radius}. Answer in " \7            "joules."8  given height: "400 km"9  given satellite: "200 kg"10  given earth: "6.0e24 kg"11  given radius: "6.4e6 m"12  asks energy: "How much energy must be expended to rocket the satellite out of the earth's gravitational influence"13end

solution.rb

1require_relative "../questions/a_satellite_rocketed_free"2require_relative "../science/gravitation/earth_satellites"34solve :a_satellite_rocketed_free, using: EarthSatellites do5  objects :satellite6  paths :orbit7  satellite.goes(:round, on: orbit)89  take height: orbit.height_of_the_orbit10  take satellite: satellite.mass11  take earth: orbit.central_mass12  take radius: orbit.central_radius1314  energy = satellite.binding_energy15  answer energy:, in: "J", figures: 21617  conclusion "In orbit the satellite's kinetic energy makes up half of its negative potential energy, so it is " \18             "still bound; freeing it takes the other half."19end

playground.rb

1require_relative "../solutions/a_satellite_rocketed_free"23play_with :a_satellite_rocketed_free do4  title "A Satellite Rocketed Free"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	UniformCircularMotion
gravitational constant	$G$	N m² kg⁻²	—	UniversalGravitation
mass	$m$	kg	0 … ∞	UniversalGravitation
radius of the body	$R$	m	0 … ∞	UniversalGravitation
surface gravity	$g_{s}$	m/s²	—	UniversalGravitation
separation	$r_{g}$	m	0 … ∞	UniversalGravitation
gravitational intensity	$E_{g}$	m/s²	—	UniversalGravitation
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
radius	$r$	m	0 … ∞	UniformCircularMotion
speed	$v$	m/s	0 … ∞	UniformCircularMotion
angular speed	$ω$	rad/s	—	UniformCircularMotion
revolutions	$n_{rev}$	—	—	UniformCircularMotion
frequency	$ν$	Hz	0 … ∞	UniformCircularMotion
centripetal acceleration	$a_{c}$	m/s²	—	UniformCircularMotion
against gravity	$q$	—	—	UniformCircularMotion
period	$T$	s	0 … ∞	UniformCircularMotion
gravity	$g$	m/s²	—	UniformCircularMotion
centripetal force	$F_{c}$	N	—	UniformCircularMotion
central mass	$M$	kg	—	EarthSatellites
central radius	$R_{c}$	m	—	EarthSatellites
height of the orbit	$h$	m	—	EarthSatellites
keplers constant	$k_{K}$	s²/m³	—	EarthSatellites
binding energy	$E_{b}$	J	—	EarthSatellites
displacement of satellite during round	$Δ x$	m	—
position of satellite at end of round	$x_{1}$	m	—
position of satellite at start of round	$x_{0}$	m	—
average velocity of satellite during round	$v_{av}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of satellite during round	$c_{av}$	m/s	—
path length of satellite during round	$d$	m	—
speed of satellite during round	$v$	m/s	0 … ∞
angular speed of satellite during round	$ω$	rad/s	—
radius of orbit	$r_{o}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of satellite during round	$ν$	Hz	0 … ∞
revolutions of satellite during round	$n_{rev}$	—	—
centripetal acceleration of satellite during round	$a_{c}$	m/s²	—
against gravity of satellite during round	$q$	—	—
centripetal force on satellite during round	$F_{c}$	N	—
mass of satellite	$m$	kg	0 … ∞
central mass of orbit	$M_{o}$	kg	—
keplers constant of orbit	$k_{K,o}$	s²/m³	—
central radius of orbit	$R_{c,o}$	m	—
height of the orbit of orbit	$h_{o}$	m	—
binding energy of satellite during round	$E_{b}$	J	—
surface gravity of ground	$g_{s,g}$	m/s²	—
mass of ground	$m_{g}$	kg	0 … ∞
radius of the body of ground	$R_{g}$	m	0 … ∞
surface gravity of satellite	$g_{s}$	m/s²	—
radius of the body of satellite	$R$	m	0 … ∞
gravitational intensity of ground by satellite	$E_{g,g}$	m/s²	—
separation of ground from satellite	$r_{g,g}$	m	0 … ∞
gravitational intensity of satellite by ground	$E_{g,g}$	m/s²	—
separation of satellite from ground	$r_{g}$	m	0 … ∞

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

starting position

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

final position

g_{s,b} = \frac{G \cdot m_{b}}{{R_{b}}^{2}}

gravity at the surface

E_{g,oo} = \frac{G \cdot m_{o}}{{r_{g,oo}}^{2}}

law of gravitation

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

Δ x = x_{1} - x_{0}

change of position(round)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(round)

c_{av} = \frac{d}{Δ t}

path a time(round)

v = ω \cdot r_{o}

speed round the circle(round)

ω = 2 \cdot \frac{π}{T}

once round a period(round)

ν = \frac{1}{T}

rounds a second(round)

ν = \frac{n_{rev}}{Δ t}

rounds over the time(round)

a_{c} = \frac{v^{2}}{r_{o}}

towards the centre(round)

q = \frac{a_{c}}{g}

measured in gs(round)

F_{c} = m \cdot a_{c}

mass times the turning(round)

v^{2} = \frac{G \cdot M_{o}}{r_{o}}

held in orbit by gravity(round)

k_{K,o} = \frac{4 \cdot π^{2}}{G \cdot M_{o}}

keplers constant of an orbit(round)

T^{2} = k_{K,o} \cdot {r_{o}}^{3}

law of periods(round)

r_{o} = R_{c,o} + h_{o}

orbit radius from the centre(round)

E_{b} = \frac{G \cdot M_{o} \cdot m}{2 \cdot r_{o}}

binding energy of an orbit(round)

g_{s,g} = \frac{G \cdot m_{g}}{{R_{g}}^{2}}

gravity at the surface(ground)

g_{s} = \frac{G \cdot m}{R^{2}}

gravity at the surface(satellite)

E_{g,g} = \frac{G \cdot m}{{r_{g,g}}^{2}}

law of gravitation(ground, satellite)

E_{g,g} = \frac{G \cdot m_{g}}{{r_{g}}^{2}}

law of gravitation(satellite, ground)