A Satellite Rocketed Free

Q. A satellite orbits the earth at a height of 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; mass of the earth = 6.0 × 10²⁴ kg; radius of the earth = 6.4 × 10⁶ m. Answer in joules. expects 5.9 × 10⁹ J

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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.

The playground

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}

The solution

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 Ruby

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

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

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

The laws

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)